Systems and Methods for Communicating by Modulating Data on Zeros in the Presence of Channel Impairments

ABSTRACT

Communication systems and methods in accordance with various embodiments of the invention utilize modulation on zeros. Carrier frequency offsets (CFO) can result in an unknown rotation of all zeros of a received signal&#39;s z-transform. Therefore, a binary MOCZ scheme (BMOCZ) can be utilized in which the modulated binary data is encoded using a cycling register code (e.g. CPC or ACPC), enabling receivers to determine cyclic shifts in the BMOCZ symbol resulting from a CFO. Receivers in accordance with several embodiments of the invention include decoders capable of decoding information bits from received discrete-time baseband signals by: estimating a timing offset for the received signal; determining a plurality of zeros of a z-transform of the received symbol; identifying zeros from the plurality of zeros that encode received bits by correcting fractional rotations resulting from the CFO; and decoding information bits based upon the received bits using a cycling register code.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present invention claims priority to U.S. Provisional PatentApplication Ser. No. 62/802,578 entitled “Robust Transceiver Designs forFrequency and Time Offsets”, filed Feb. 7, 2019, the disclosures ofwhich is herein incorporated by reference in its entirety.

FIELD OF THE INVENTION

The disclosure relates generally to communication systems and morespecifically to blind communication schemes that transmit over unknownwireless multipath channels.

BACKGROUND

The future generation of wireless networks faces a diversity of newchallenges. Trends on the horizon—such as the emergence of the Internetof Things (IoT) and the tactile Internet—have radically changed thinkingabout how to scale wireless infrastructure. Among the main challengesnew emerging technologies have to cope with is the support of a massivenumber (billions) of devices ranging from powerful smartphones andtablet computers to small and low-cost sensor nodes. These devices comewith diverse and even contradicting types of traffic including highspeed cellular links, massive amounts of machine-to-machine (M2M)connections, and wireless links which carrying data in short-packets.Although intensively discussed in the research community, the mostfundamental question of how we will communicate in the near future undersuch diverse requirements remains largely unresolved.

A key problem of supporting sporadic and short-message traffic types ishow to acquire, communicate, and process channel information.Conventional channel estimation procedures typically require asubstantial amount of resources and overhead. This overhead can dominatethe intended information exchange when the message is short and thetraffic sporadic. For example, once a node wakes up in a sporadic mannerto deliver a message it has first to indicate its presence to thenetwork. Secondly, training symbols (pilots) are typically used toprovide sufficient information at the receiver for estimating linkparameters such as the channel coefficients. Finally, after exchanging acertain amount of control information, the device transmits its desiredinformation message on pre-assigned resources. In current systems thesesteps are usually performed sequentially in separate communicationphases yielding a tremendous overhead once the information message issufficiently short and the nodes wake up in an unpredictable way.Therefore, a redesign and rethinking of several well-established systemconcepts and dimensioning of communication layers will likely benecessary to support such traffic types in an efficient manner.Noncoherent and blind strategies, provide a potential way out of thisdilemma. Classical approaches like blind equalization have beeninvestigated in the engineering literature, but new noncoherentmodulation ideas which explicitly account for the short-message andsporadic type of data will likely be required.

In many wireless communication scenarios the transmitted signals areaffected by multipath propagation and the channel will therefore befrequency-selective if the channel delay spread exceeds the samplingperiod. Additionally, in mobile and time-varying scenarios oneencounters also time-selective fast fading. In both cases channelparameters typically have a random flavor and potentially cause variouskinds of interference. From a signal processing perspective it is,therefore, desirable to take care of possible signal distortions, at thereceiver and potentially also at the transmitter.

A known approach to deal with multipath channels is to modulate data onmultiple parallel waveforms, which are well-suited for the particularchannel conditions. A simple approach that can be utilized forfrequency-selective multipath channels is orthogonal frequency divisionmultiplexing (OFDM). When the maximal channel delay spread is known,inter-symbol-interference (ISI) can be avoided by a suitable guardinterval. Orthogonality of the subcarriers can be achieved by a cyclicprefix preventing inter-carrier-interference. On the other hand, from aninformation-theoretic perspective, random channel parameters are helpfulfrom a diversity view point. Spreading data over subcarriers can exploitdiversity in a frequency-selective fading channel. But to coherentlydemodulate the data symbols at the receiver, the channel impulseresponse (CIR) should be known at least at the receiver. To gainknowledge of the CIR, training data (pilots) are typically added to theinformation baseband samples, leading to a substantial overhead when thenumber of samples per signal is in the order of the channel taps. If thenumber of samples is even less than the number of channel taps, it canbe mathematically impossible to accurately estimate from any pilot datathe channel (assuming full support). Hence, one is either forced toincrease the signal length by adding more pilots or assume someside-information on the channel. Furthermore, the pilot density has tobe adapted to the mobility and, in particular, OFDM is very sensitive totime-varying distortions due to Doppler shift and oscillatorinstabilities. Dense CIR updates are often required, which can result incomplex transceiver designs.

Due to ubiquitous impairments between transmitter and receiver clocks acarrier frequency offset (CFO) is likely to be present after a downconversion to the baseband. Doppler shifts due to relative velocity canalso cause additional frequency dispersion which can be approximated infirst order by a CFO. This is a known weakness in many multi-carriermodulation schemes, such as OFDM, and various approaches have beendeveloped to estimate or eliminate the CFO effect. A common approach forOFDM systems is to learn the CFO in a training phase or from blindestimation algorithms, such as MUSIC or ESPRIT. Furthermore, due to theunknown distance and asynchronous transmission, a timing offset (TO) ofthe received symbol typically has to be determined as well, which willotherwise destroy the orthogonality of the OFDM symbols. By“sandwiching” the data symbol between training symbols a timing andfrequency offset can be estimated. By using antenna arrays at thereceiver, antenna diversity of a single-input-multiple output (SIMO)system can be exploited to improve the performance.

OFDM is typically used in long packets (frames), consisting of manysuccessive symbols resulting in long signal lengths. In a sporadiccommunication, only one packet is transmitted and the next packet mayfollow at an unknown time later. In a random access channel, a differentuser may transmit the next packet from a different location. Such apacket experiences an independent channel realization. Hence, thereceiver can barely use any channel information learned from the past.To reduce overhead and power consumption, again low-latency andshort-packet durations are favorable.

A communication system that supports sporadic packet communication atultra low latency can be useful in a number of applications including(but not limited to) critical control applications, like commands inwireless for high performance (HP), ad hoc signaling protocols, secretkey authentication, or initiation, synchronization and channel probingpackets to prepare for longer or future transmission phases. At the sametime, ultra reliability is required for such low latency communications(URLLC), especially if dealing with industrial wireless scenarios wherethe packets contain critical control data. The IEEE 802.11 standard forWirelessHP specifies a target packet error probability of 10⁻⁹ atscheduling units (SU) below 1 μs for power system automation and powerelectronic control. Here, the SU is the actual transmission time betweentransmitter and receiver. Furthermore, the next generation of mobilewireless networks aims for large bandwidths with carrier frequenciesbeyond 30 GHz, in the so called mmWave band, such that the samplingperiod is in the order of nano seconds. Hence, even at moderatemobility, the wireless channel remains approximately time-invariant onlyfor a short duration, encouraging shorter symbol lengths. On the otherhand, wideband channels resolve many multipaths due to the largebandwidth, which makes equalizing in the time-domain very challengingand is commonly simplified by using OFDM instead. But this can requirelong signal lengths, due to additional pilots needed to learn thechannel. This may not be feasible if the transmission time or schedulingunit requirement is too short. Also, the maximal CIR length needs to beknown at the transmitter and if underestimated might lead to a seriousperformance loss. Considering indoor channels at 60 GHz, a delay spreadof up to 30 ns can still be present. Considering a bandwidth of 1 GHzresults in a CIR length of 30 and a signal length of 100-1000 to meetthe SU requirement. Hence, exploiting multiple OFDM symbols becomes moreand more challenging if the receive duration approaches the order of thechannel delay spread. Therefore, one-shot symbol transmissions canbecome necessary to push the latency on the physical-layer to itsphysical limits.

Another issue in transmitting at high frequencies can be attenuation,which can be overcome by exploiting beam-forming with massive antennaarrays. For massive or mobile users, this again increases the complexityand energy consumption for estimating and tracking the huge amount ofchannel parameters. Furthermore, a bottleneck in mmWave MIMO systems canbe the blockage of line-of-sight connections which require wider ormultiple beams, resulting in a significant reduction of the receivepower.

SUMMARY OF THE INVENTION

Systems and methods in accordance with many embodiments of the inventionutilize timing-offset (TO) and carrier frequency offset (CFO) estimationin a BMOCZ scheme to communicate over unknown multipath channelsincluding (but not limited to) wideband frequency-selective fadingchannels. In several embodiments, the CFO robustness is realized by acyclically permutable code (CPC), which can enable a receiver toidentify the CFO. In a number of embodiments, an oversampled DiZeTdecoder is utilized within the receiver that is capable of estimatingthe CFO. In certain embodiments, CPC codes are utilized that areconstructed with cyclic BCH codes, which can provide the capability tocorrect additional bit errors and can enhance the performance of theBMOCZ scheme for moderate SNRs.

Due to the low-latency of BMOCZ the CFO and TO estimation can beperformed from as few as one single BMOCZ symbol. This blind scheme canbe ideal for control-channel applications, where a limited amount ofcritical and control data is exchanged while at the same time, channeland impairments information needs to be communicated and estimated.Communication systems that employ BMOCZ coded with a CPC in accordancewith many embodiments of the invention can enable low-latency andultra-reliable short-packet communications over unknown widebandchannels.

One embodiment of the communication system includes a transmitterhaving: an encoder configured to receive a plurality of information bitsand output a plurality of encoded bits in accordance with a cyclingregister code (CRC); a modulator configured to modulate the plurality ofencoded bits to obtain a discrete-time baseband signal, where theplurality of encoded bits are encoded in the zeros of the z-transform ofthe discrete-time baseband signal; and a signal generator configured togenerate a continuous-time transmitted signal based upon thediscrete-time baseband signal. In addition, the communication systemincludes a receiver, having: a demodulator configured to down convertand sample a received continuous-time signal at a given sampling rate toobtain a received discrete-time baseband signal, where the receiveddiscrete-time baseband signal includes at least one of a timing offset.(TO) and a carrier frequency offset (CFO); and a decoder configured todecode a plurality of bits of information from the receiveddiscrete-time baseband signal. Furthermore, the decoder decodes theplurality of bits of information by: estimating a TO for the receiveddiscrete-time baseband signal to identify a received symbol; determininga plurality of zeros of a z-transform of the received symbol;identifying zeros from the plurality of zeros that encode a plurality ofreceived hits; and decoding a plurality of information hits based uponthe plurality of received bits using the CRC.

In a further embodiment, the receiver receives the continuous-timetransmitted signal over a multipath channel.

In another embodiment, the modulator is configured to modulate theplurality of encoded bits so that the z-transform of the discrete-timebaseband signal comprises a zero for each of a plurality of encodedbits.

In a still further embodiment, the modulator is configured to modulatethe plurality of encoded bits so that each zero in the z-transform ofthe discrete-time baseband signal is limited to being one of a set ofconjugate-reciprocal pairs of zeros.

In still another embodiment, each conjugate reciprocal pair of zeros inthe set of conjugate-reciprocal pairs of zeros comprises: an outer zerohaving a first radius that is greater than one; and an inner zero havinga radius that is the reciprocal of the first radius. In addition, theinner and outer zero have phases that are the same phase, the radii ofthe outer zeros in each pair of zeros in the set of conjugate-reciprocalpairs of zeros are the same, and the phases of the outer zeros in eachpair of zeros in the set of conjugate-reciprocal pairs of zeros areevenly spaced over one complete revolution.

In a yet further embodiment, the cycling register code is a cyclicallypermutable code (CPC).

In yet another embodiment, the CPC is extracted from a Bose ChaudhuriHocquenghem (BCH) code.

In a further additional embodiment, the CPC is extracted form aprimitive BCH code.

In another additional embodiment, the CPC has a code length that is aMersenne prime.

In a further embodiment again, the CPC has a code length selected fromthe group consisting of 3, 7, 31, and 127.

In another embodiment again, the CRC is generated by an inner code andan outer code which are combined in a non-linear fashion.

In a still yet further embodiment, the outer code is a cycling registercode having a lower code rate than the inner code.

In still yet another embodiment, the outer code is a cyclicallypermutable code (CPC).

In a still further additional embodiment, the CRC is an affine CPC(ACPC) code.

In still another additional embodiment, the ACPC is characterized bybeing attainable using a cyclic inner code having codewords of an innercodeword length, which is affine translated by a given binary word ofthe inner codeword length, and then further encoded by a cyclic outercode.

In a still further embodiment again, the decoder is configured toestimate the timing offset by measuring energy over an expected symbollength with a sliding window in the sampled signal.

In still another embodiment again, the decoder is configured to measureenergy over an expected symbol length by convolving samples with auniversal Huffman sequence of the expected symbol length comprising twoimpulses at the beginning and the end of the expected symbol length.

In a yet further additional embodiment, the decoder is configured toestimate the TO by identifying a set of three energy peaks that yield amaximum energy sum over an expected symbol length.

In yet another additional embodiment, the demodulator is configured tooversample the received continuous-time signal, and the decoder isconfigured to identify zeros from the plurality of zeros that encode aplurality of received bits by identifying a fractional rotationresulting from the carrier frequency offset.

In a yet further embodiment again, the decoder is configured todetermine a most likely set of zeros for the z-transform of thediscrete-time baseband signal used to generate the transmitted signalbased upon the received symbol.

In yet another embodiment again, the decoder is configured to determinethe plurality of received bits by performing a weighted comparison ofsamples of the z-transform of the received symbol with each zero in aset of zeros.

In a further additional embodiment again, each zero in the z-transformof the discrete-time baseband signal used to generate the transmittedsignal is limited to being one of a set of conjugate-reciprocal pairs ofzeros.

In another additional embodiment again, the receiver comprises aplurality of receive antennas and the decoder determines the pluralityof information bits by combining values derived from the samples of aplurality of continuous-time signals received by the plurality ofreceive antennas to perform decoding.

A transmitter in accordance with one embodiment of the inventionincludes: an encoder configured to receive a plurality of informationbits and output a plurality of encoded bits in accordance with a cyclingregister code (CRC); a modulator configured to modulate the plurality ofencoded bits to obtain a discrete-time baseband signal, where theplurality of encoded bits are encoded in the zeros of the z-transform ofthe discrete-time baseband signal; and a signal generator configured togenerate a continuous-time transmitted signal based upon thediscrete-time baseband signal.

In a further embodiment, the continuous-time transmitted signalcomprises a carrier frequency modulated based upon the discrete-timebaseband signal.

A receiver in accordance with one embodiment of the invention includes ademodulator configured to down convert, and sample a receivedcontinuous-time signal to obtain a received discrete-time basebandsignal and over sample the received discrete-time signal by zeropadding, where the received discrete-time baseband signal includes atleast one of a timing offset (TO) and a carrier frequency offset (CFO);and a decoder configured to decode a plurality of bits of informationfrom the received discrete-time baseband signal by: estimating a TO forthe received discrete-time baseband signal to identify a receivedsymbol; determining a plurality of zeros of a z-transform of thereceived symbol; identifying zeros from the plurality of zeros thatencode a plurality of received bits by identifying and correcting afractional rotation in the plurality of zeros resulting from the CFO;and decoding a plurality of information bits based upon the plurality ofreceived bits using a cycling register code (CRC).

BRIEF DESCRIPTION OF THE DRAWINGS

It should be noted that the patent or application file contains at leastone drawing executed in color. Copies of this patent or patentapplication publication with color drawing(s) will be provided by theOffice upon request and payment of the necessary fee.

FIG. 1 conceptually illustrates multipath propagation of transmittedsignals in a wireless channel.

FIG. 2 conceptually illustrates a transmitter and receiver that utilizea binary MOCZ scheme over an unknown multipath channel distorted byadditive noise implemented in accordance with an embodiment of theinvention.

FIG. 3 conceptually illustrates carrier frequency-offset (CFO)estimation via oversampling the DiZeT decoder for K=6 data zeros andwithout channel zeros by a factor of Q=K=6 in accordance with anembodiment of the invention. Blue circles denote the codebook

(6), solid blue circles the transmitted zeros α_(k) and red the receivedrotated zeros {tilde over (α)}_(k). The conjugate-reciprocal zero pairsare pairwise separated by the basis angle θ₆.

FIG. 4A shows the magnitudes of a Huffman sequence (BMOCZ symbol) forK=127, as pictured by blue circles, and a Huffman bracket or universalHuffman sequence, as pictured with red crosses.

FIG. 4B shows the autocorrelation magnitudes as well as the correlationbetween the Huffman and bracket sequences of FIG. 4A.

FIG. 5A shows maximal interior Huffman sample energies for K∈{6, 7, 10}over η.

FIG. 5B shows probability of wrong timing offset detection over SNR forideal AWGN channels and multipath with L=10, LOS path and p=0.98 withrandom CFO for 5·10⁵ runs per SNR point and N_(obs)=270 for K∈{1, 4, 5,127} with R_(uni)(K) in equation (8) below.

FIG. 6 shows an iterative back stepping algorithm for estimating timingoffset in accordance with an embodiment of the invention.

FIG. 7A shows the absolute-square (energy) of the CIR samples (redcrosses) and of the samples of a BMOCZ symbol (blue circles). The CIRexperience only NLOS paths with length L=127 and S=83 active paths at anaverage power decay of p^(n)=0.95^(n) for the nth delay. The normalizedBMOCZ symbol has K=127 zeros with 62 outside the unit circle (bit-valueone).

FIG. 7B shows the corresponding received energy samples (blue-circles)at SNR=18 dB. An approximation of the CIR energy samples (red-crosses)|h_(n)′|² can be observed as an echo of the last BMOCZ coefficient. Theefficient CIR length is determined at L_(eff)=81 at a timing offset ofτ=77 with strongest path at τ_(s)=92.

FIG. 8 shows an algorithm for estimating channel length based uponchannel energy in accordance with an embodiment of the invention.

FIG. 9 is a chart showing bit-error-rate (BER) over rSNR for BMOCZ withDiZeT for random ϕ∈[0, θ_(K)/2) for L=K=32.

FIG. 10A shows simulated results for a SIMO system for M=1, 2, 4 receiveantennas over 2000 runs per point at p=0.95 and S=42. In addition,results are presented for a BCH code with no carrier frequency offsetand timing offset and an ACPC code with a carrier frequency offset andtiming offset using the CIRED Algorithm and the CPBS algorithm.

FIG. 10B shows timing offset and carrier frequency offset estimationerrors for M=1, 2 and M=4 receive antennas in the simulations shown inFIG. 10A.

FIG. 11 shows a comparison to two successive OFDM-DPSK blocks, oneOFDM-IM, a pilot and QPSK, and a BMOCZ block in time-domain. The solidbars denote the used subcarriers/zeros.

FIGS. 12A and 12B shows BER respectively block-error-rate (BLER)simulation results for arbitrary carrier frequency with an ACPC-(31,11)code for B=11 bits. The E_(b)/N₀ averaged over the fading channel isdefined here as 1/BN₀ for B information bits.

FIG. 13 shows BER over E_(b)/N₀ with M=2 and 4 antennas forBMOCZ-ACPC-(31,16) distorted by timing and carrier frequency offsets(green curves). OFDM-IM is shown without carrier frequency offsets andtiming offsets.

FIG. 14A shows simulated BER of blind schemes over instantaneous SNRcorrected by spectral efficiency.

FIG. 14B shows random channel realizations with Quadriga simulations atmaximal delay spread L=41.

FIG. 15A shows a comparison of BMOCZ to OFDM-IM, -GIM, -DPSK with 2 OFDMsymbols, and Pilot-QPSK for 1 and 4 antennas at maximal CIR length L=9.

FIG. 15B shows a comparison of BMOCZ to OFDM-IM, -GIM, -DPSK with 2 OFDMsymbols, and Pilot-QPSK for 1 and 4 antennas at maximal CIR length L=4.

FIG. 16 shows MATLAB code for creating the generator and check matricesof an ACPC-(n, k−m) code, as well as the binary codeword for the affineshift in accordance with an embodiment of the invention.

DETAILED DESCRIPTION

Turning now to the drawings, communication systems and methods inaccordance with various embodiments of the invention utilize a blind(noncoherent) communication scheme, called modulation onconjugate-reciprocal zeros (MOCZ), which can reliably transmit sporadicshort-packets of fixed size over channels including (but not limited to)unknown wireless multipath channels. In many embodiments, communicationsystems utilize MOCZ schemes to transmit reliable and robust data withas few as one symbol in the presence of timing and/or frequency offsets.As is discussed further below, a carrier frequency offset (CFO) canresult in an unknown rotation of all zeros of a received signal'sz-transform. Therefore, in several embodiments, a binary MOCZ scheme(BMOCZ) is utilized in which the modulated binary data is encoded usinga cyclically permutable code (CPC), which can cause the BMOCZ symbol tobe invariant with respect to any cyclic shift resulting from a CFO. In anumber of embodiments, the communication systems can utilize techniquessimilar to those described herein to achieve high spectral efficiencyand/or low-latency.

In several embodiments, a transmitter modulates the information of apacket on the zeros of a transmitted discrete-time baseband signal'sz-transform. The discrete-time baseband signal can be called a MOCZsymbol, which is a finite length sequence of complex-valuedcoefficients. These coefficients can then modulate a continuous-timepulse shape at a sample period of T=1/W, where W is the bandwidth, togenerate a continuous-time baseband waveform. Since the MOCZ symbols(sequences) are neither orthogonal in the time nor frequency domain, theMOCZ design can be seen as a non-orthogonal multiplexing scheme. Afterup-converting to the desired carrier frequency by the transmitter, thetransmitted passband signal can propagate in space such that, due toreflections, diffractions, and/or scattering, different, delays of theattenuated signal can interfere at the receiver. Hence, multipathpropagation can cause a time-dispersion which can result in afrequency-selective fading channel. Furthermore, a CFO is likely to bepresent after a down conversion to baseband at the receiver.

Unlike OFDM, which is typically used in long frames, MOCZ can beutilized with shorter transmissions. In a bursty signaling scheme,timing and carrier frequency offsets may need to be addressed from onlyone received symbol or a small number of received symbols. In MOCZschemes in accordance with a number of embodiments of the invention,communication is scheduled and timed on a MAC layer by a certain bus,running with a known bus clock-rate. Therefore, timing-offsets of thesymbols can be assumed as fractions of the bus clock-rate. Accordingly,MOCZ receivers in accordance with many embodiments of the invention canbe designed to be robust to these impairments.

In a communication system that employs MOCZ, a CFO is likely to resultin an unknown common rotation of all received zeros. Since the angularzero spacing in a BMOCZ symbol of length K+1 is given by a base angle of2π/K, a fractional rotation can be easily obtained at the receiver by anoversampling during the post-processing to identify the most likelytransmitted zeros (zero-pattern).

Rotations, which are integer multiples of the base angle, can correspondto cyclic shifts of the binary message word. By using a CPC for thebinary message, which can be extracted from cyclic codes, such as (butnot limited to) Bose Chaudhuri Hocquenghen (BCH) code, a BMOCZ symbolcan become invariant against cyclic shifts and hence against, a CFOpresent with a communication system. In several embodiments, use of aCPC to encode a binary message can enable the operation of acommunication system in a manner that does not rely upon symboltransmissions for estimation of a CFO, which can reduce overhead,latency, and complexity. In a number of embodiments, the communicationsystem is able to provide a CFO estimation as part of the decodingprocess of a single BMOCZ symbol. In a number of embodiments, thereceiver is able to estimate the cyclic shift that has been introducedby the CFO based upon the mapping of the zeros of the z-transform of thereceived symbol to bits and the CPC used to encode the transmittedmessage. Furthermore, the error correction capabilities of the CPC canalso improve the BER and moreover the block error-me (BLER) performanceof the communication system tremendously.

By measuring the energy of the expected symbol length with a slidingwindow in the received signal, a receiver in accordance with variousembodiments of the invention can identify arbitrary TOs. The robustnessof the TO estimation can be demonstrated analytically, which revealsanother strong property of communication systems that employ MOCZschemes. Once the TO is determined, the symbol can be decoded and theCPC utilized to correct for any CFO.

In many embodiments, communication systems employ multiple receiveantennas and achieve robustness to CFO and TO using a CPC for errorcorrection. By simulating BER over the received SNR for various averagepower delay profiles, with constant and exponential decay as well asrandom sparsity constraints, performance of communication systems inaccordance with various embodiments of the invention in different indoorand outdoor scenarios can be demonstrated.

Communication systems, transmitters, and receivers that can be utilizedto transmit data encoded using a CPC by MOCZ in accordance with variousembodiments of the invention are discussed further below.

Notation

In the discussion that follows, small letters are used for complexnumbers in

. Capital Latin letters denote natural numbers

and refer to fixed dimensions, where small letters are used as indices.Boldface small letters denote row vectors and capitalized letters referto matrices. Upright capital letters denote complex-valued polynomialsin

[z]. The first N natural numbers in

are denoted as [N]: {0, 1, . . . , N−1}. For K∈

, K+[N]={K, K+1, . . . K+N−1} is denoted by the K-shift of the set [N].The Kronecker-delta symbol is given by δ_(nm) and is 1 if n=m and 0otherwise. For a complex number x=a+jb, given by its real part Re(x)=a∈

and imaginary part Im(x)=b∈

with imaginary unit j=√{square root over (−1)}, its complex-conjugationis given by x=a−jb and its absolute value by

${x} = {\sqrt{x\;\overset{\_}{x}}.}$

For a vector x∈

^(N) its complex-conjugated time-reversal or conjugate-reciprocal isdenoted by

, given as

The discussion also uses A*=Ā^(T) for the complex-conjugated transposeof the matrix A. The N×N identity is written as I_(N) and a N×M matrixwith all elements zero is written as O_(N,M). D_(x) refers to thediagonal matrix generated by x∈

^(N). The N×N unitary Fourier matrix F=F_(N) is given entry-wise byf_(l,k)=e^(−j2πlk/N)/√{square root over (N)} for l, k∈[N]. T∈

^(N×M) denoted the elementary Toeplitz matrix given element-wise asδ_(n-lm). The all one and all zero vectors in dimension N will bedenoted by 1_(N) and 0_(N), respectively. The Euclidean basis vectors in

^(N) are given by {e₁, . . . , e_(N)} elementwise as (e_(i))_(j)=δ_(i,j)for i, j∈1+[N]. The

_(p)-norm of a vector x=(x₀, . . . , x_(N-1))∈

^(N) is given by ∥x∥_(p)=(Σ_(k=0) ^(N-1)|x_(k)|^(p))^(1/p) for p≥1. Ifp=∞, then ∥x∥_(∞)=max_(k)|x_(k)| and for p=2∥x∥₂=∥x∥. Finally, theexpectation of a random variable x is denoted by

[x].

System Model

Systems and methods in accordance with many embodiments of the inventioninvolve transmission of data of an unknown multipath channel. Multipathpropagation of transmitted data is conceptually illustrated in FIG. 1.In the illustrated embodiment, the communication system 100 includes abase station 102 that is acting as a broadcasting transmitter and anarbitrary mobile station 104 that is acting as a receiver. In theillustrated embodiment, the signal transmitted by the base station 102can be received by the mobile station 104 via six different 106 paths ofpropagation. Therefore, the signal received by mobile station 104 is asuperposition of the signals received via the different paths subject tothe various delays experienced by the signals. The channel alsointroduces noise. In the illustrated embodiment, the message data isencoded using a CPC to provide robustness to CFO and/or TO. As isdiscussed further below, CPCs are just one example of a class ofappropriate code and the various embodiments described herein should beunderstood as not being limited to a single class of code. Instead,systems and methods in accordance with various embodiments of theinvention can utilize any code that enables a receiver to detect and/orcorrect a cyclic rotation (permutation) of the transmitted message bitsas appropriate to the requirements of specific applications.

A transmitter and receiver that utilize a binary MOCZ scheme over anunknown multipath channel implemented in accordance with an embodimentof the invention are conceptually illustrated in FIG. 2. Thecommunication system 200 includes a modulator within the transmitter 202that performs a modulation operation f to transmit data over a multipathchannel 204 and a decoder 206 within a receiver that performs ademodulation/decoder operation g. As discussed below, both themodulation operation f and the decoder operation g rely on at least onshared zero-codebook

. It is in principle possible that a receiver can blindly identify theused zero-codebook taken from a predefined set of zero-codebooks toallow for example multiple access schemes. As is discussed below, thetransmitter and receiver described herein are practical and can beimplemented using logic devices including (but not limited to) fieldprogrammable gate arrays, digital signal processors, and or any of avariety of conventional software defined radio platforms.

The binary message sequence m_(k)∈{0, 1} can be chunked at thetransmitter in blocks of length K. As is discussed further below, theseblocks can then be encoded using a code that is robust to cyclicpermutations. In several embodiments, the encoding can involve multiplecodes. In certain embodiments, a combination of an inner code an outercode is utilized, where the inner code is a cyclic code which is affinetranslated and an outer code which is a cyclic code that provides errorcorrection capabilities and robustness against cyclic shifts. As canreadily be appreciated the specific code and/or combination of codesthat are utilizes within a transmitter are largely dependent upon therequirements of specific applications in accordance with variousembodiments of the invention.

For BMOCZ, the modulator f encodes the block m to a normalizedcomplex-valued symbol (sequence) x=(x₀, . . . , x_(K+1))^(T)∈

^(K+1+1) by using the zero-codebook

of cardinality 2^(K). In many embodiments, when the block size K issmall, the discrete-time BMOCZ symbols can be pre-generated using themethods described below for creating a codebook, and is selected using alookup mechanism such as (but not limited to) a lookup table. The BMOCZsymbol x is typically modulated onto a carrier frequency f_(c) with apulse generator running at a sampling clock T=1/W to transmit areal-valued passband signal of bandwidth W to the receiver over anunknown time-invariant channel with a maximum delay spread ofT_(d)=(L−1)T which resolves L equally spaced multi-paths (delays). Aftera down-converting and sampling to the discrete-time baseband, thereceiver observes the channel output y under an unknown additive noisevector w. As noted above, the down conversion can also introduce a CFOand/or a TO. The demodulator/decoder 206 can obtain from the receivedsignal an estimated block codeword {circumflex over (m)} by theknowledge of the zero-codebook

. As is discussed further below, use of a CPC in the encoding of thetransmitted data enables the receiver to decode the transmitted datadespite the presence of a CFO and/or a TO.

Although a specific binary MOCZ scheme is described above with referenceto FIG. 2, as is discussed below, a variety of MOCZ schemes can beimplemented as appropriate to the requirements of a given application inaccordance with various embodiments of the invention including (but notlimited to) communication systems that employ M-ary MOZ schemes, MOCZschemes, multiple receive antennas and/or outer codes to provideadditional error correction capabilities enabling recovery of messagebits in the face of received bit errors. In order to appreciate thesedifferent variants, a more complex and generalized model of acommunication system that utilizes a MOCZ scheme is described below.

System Model and Requirements

Communication systems and methods in accordance with many embodiments ofthe system utilize a blind and asynchronous transmission of a shortsingle MOCZ symbol at a designated bandwidth W. In this “one-shot”communication, it can be assumed that no synchronization and no packetscheduling occurs between the transmitter and receiver. Such extremesporadic, asynchronous, and ultra short-packet transmissions are oftenemployed in applications that include (but are not limited to): criticalcontrol applications, exchange of channel state information (CSI),signaling protocols, secret keys, authentication, commands in wirelessindustry applications, or initiation, and synchronization and channelprobing packets to prepare for longer or future transmission phases.

By choosing the carrier frequency, transmit sequence length, andbandwidth accordingly, a receive duration in the order of the channeldelay spread can be obtained, which can reduce latency at the receiver(potentially to the lowest latency possible). Since the next generationof mobile wireless networks aims for large bandwidths with carrierfrequencies beyond 10 Ghz, in the so called mm Wave band, thetransmitted signal duration of a signal transmitted in accordance withan embodiment of the invention can be in the order of nano seconds.Hence, even at moderate mobility, the wireless channel in an indoor oroutdoor scenario can be considered as approximately time-invariant oversuch a short time duration. On the other hand, wideband channels can behighly frequency selective, which can be due to the superposition ofdifferent delayed versions (echos) of the transmitted signal at thereceiver. This can make performing equalization in the time-domain verychallenging and is commonly simplified by using an OFDM scheme. Butconventional OFDM typically requires an additional cyclic prefix toconvert the frequency-selective channel to parallel scalar channels and,in coherent mode, it often requires additional pilots (training) tolearn the channel coefficients. As can readily be appreciated, this canincrease latency for short messages dramatically.

For a communication in the mmWave band, massive antenna arrays can beexploited to overcome attenuation. Use of multiple receive antennas canincrease complexity and energy consumption in estimating channelparameters and can become a bottleneck in mmWave MIMO systems,especially for mobile scenarios. However, in a sporadic communicationone or a small number of symbols are transmitted and additional symbolsmay follow at an unknown later time. In a random access channel (RACH),a different, user may transmit the next symbol from a differentlocation, which will therefore experience an independent channelrealization. Hence, the receiver can barely use any channel informationlearned from past communications. OFDM systems typically approach thisby transmitting many successive OFDM symbols as a long frame, toestimate the channel impairments, which can cause considerable overheadand latency if only a few data-bits need to be communicated.Furthermore, to achieve orthogonal subcarriers in OFDM, the cyclicprefix typically has to be at least as long as the channel impulseresponse (CIR) length, resulting in signal lengths at least twice theCIR length during which the channel also needs to be static. Using OFDMsignal lengths much longer than the coherence time might not be feasiblefor fast time-varying block-fading channels. Furthermore, the maximalCIR length typically needs to be known at the transmitter and ifunderestimated can lead to a serious performance loss. This is in highcontrast to communication systems in accordance with various embodimentsof the invention that employ a MOCZ scheme, where the signal length canbe chosen for a single MOCZ symbol independently from the CIR length. Asis discussed further below, communication systems in accordance withmany embodiments of the invention encode transmitted data using CPCs toaddress the ubiquitous impairments of MOCZ schemes under such ad-hoccommunication assumptions for communications involving signal lengths inthe order of the CIR length.

After up-converting the MOCZ symbol, which is a discrete-timecomplex-valued baseband signal x=(x₀, x₁, . . . , x_(K))∈

^(K+1) of two-sided bandwidth W, to the desired carrier frequency f_(c),the transmitted passband signal will propagate in space. Regardless ofdirectional or omnidirectional antennas, the signal can be reflected anddiffracted at point-scatters, which can result in different delays ofthe attenuated signal and interfere at the receiver if the maximal delayspread T_(d) of the channel is larger than the sample period T=1/W.Hence, the multipath propagation causes time dispersion resulting in afrequency-selective fading channel. Due to ubiquitous impairmentsbetween transmitter and receiver clocks an unknown frequency offset Δfwill be present after the down-conversion to the receivedcontinuous-time baseband signal

{tilde over (r)}(t)=r(t)e ^(j2πtΔf).  (1)

A phase shift ϕ₀ can also be present after down-conversion, which canyield a received continuous-time baseband signal

{tilde over (r)}(t)=r(t)e ^(j2πtΔf+jϕ) ⁰ ,  (2)

By sampling {tilde over (r)}_(n)={tilde over (r)}(nT) at the sampleperiod T, the received discrete-time baseband signal can be representedby a tapped delay line (TDL) model. Here the channel action is given asthe convolution of the MOCZ symbol x with a finite impulse response {

}, where the

th complex-valued channel tap

describes the

th averaged path over the bin, [

T, (

+1)T), which can be modelled by a circularly symmetric Gaussian randomvariable as

$\begin{matrix}{h_{l} \in \left\{ {\begin{matrix}{{C\;{\mathcal{N}\left( {0,{s_{\ell}p^{\ell}}} \right)}},} & {l \in \lbrack L\rbrack} \\{0,} & {else}\end{matrix}.} \right.} & (3)\end{matrix}$

The average power delay profile (PDP) of the channel can be sparse andexponentially decaying, where

∈{0, 1} defines the sparsity pattern of S=|supp(h)|=

=∥s∥₁ non-zero coefficients. To obtain equal average transmit andaverage receive power the overall channel gain can be eliminated fromthe analysis by normalizing the CIR realization h=(h₀, . . . , h_(L-1))by its average energy Σ_(i=0) ^(L-1) s_(l)p^(l) (for a given sparsitypattern), such that

[∥h∥²]=1. The convolution output can then be additively distorted byGaussian noise u_(n)′∈

(0, N₀) of zero mean and variance (average power density) N₀ for n∈

as

$\begin{matrix}{{\overset{\sim}{r}}_{n} = {{e^{{{jn}\;\phi} + {j\;\phi_{0}}}\left( {{\sum\limits_{k = 0}^{K}{x_{k}h_{n - \tau_{0} - k}}} + w_{n}} \right)} = {{{\sum\limits_{k = 0}^{K}{e^{{jk}\;\phi}x_{k}e^{{{j{({n - k})}}\phi} + {j\;\phi_{0}}}h_{n - \tau_{0} - k}}} + {\overset{\sim}{w}}_{n}} = {{\sum\limits_{k = 0}^{K}{{\overset{\sim}{x}}_{k}{\overset{\sim}{h}}_{n - \tau_{0} - k}}} + {{\overset{\sim}{w}}_{n}.}}}}} & (4)\end{matrix}$

Here ϕ=2πΔf/W mod 2π∈|0, 2π) denotes the carrier frequency offset (CFO)and τ₀∈

the timing offset (TO), which marks the delay of the first symbolcoefficient x₀ via the first channel path h₀. The symbol {tilde over(x)}=xM_(ϕ)∈

^(K+1) in (4) has phase-shifted coefficients {tilde over(x)}_(k)=e^(jkϕ)x_(k) as well as the channel

${{\overset{\sim}{h}}_{\ell} = {e^{{{j{({l + r_{0}})}}\phi} + {j\;\phi_{0}}}h_{l}}},$

which will be also affected by a global phase-shift τ₀ϕ+ϕ₀. Since thechannel taps are circularly symmetric Gaussian, their phases areuniformly distributed, and shifts will not change this distribution. Bythe same argument, the noise distribution of u_(n)′ is not alternated bythe phase-shifts.

Instead of modulating the information of a message m=(m₁, . . . ,m_(K))∈[M]^(K) on time or frequency coefficients, communication systemsand methods in accordance with many embodiments of the invention canemploy a M-ary MOCZ. A MOCZ signal's z-transform is a polynomial ofdegree K that is determined by exactly K complex-valued zeros α_(k)^((m) ^(k) ⁾ and a factor x_(K) as

$\begin{matrix}{{X(z)} = {{\sum\limits_{k = 0}^{K}{x_{k}z^{k}}} = {x_{K}{\prod\limits_{k = 1}^{K}{\left( {z - \alpha_{k}^{(m_{k})}} \right).}}}}} & (5)\end{matrix}$

In several embodiments, the MOCZ encoder maps each message m of length Kto the zero-pattern (zero-constellation) α(m)=(α₁ ^((m) ¹ ⁾, . . . ,α_(K) ^((m) ^(K) ⁾)∈

^(K)=

× . . . ×

_(K)⊂

^(K) which defines iteratively

∀q=2, . . . ,K:x _(q)=(0,x _(q-1))−(α_(q) ^((m) ^(q) ⁾ x _(q-1),0) and x₁=(−α₁ ^((m) ¹ ⁾,1)  (6)

the normalized MOCZ symbol (constellation) after the last iteration stepx=x_(K)/∥x_(K)∥∈

^(K+1). Since the convolution with the CIR in (4) for the noise-freecase (no CFO, no TO) is given as a polynomial product Y(z)=X(z)H(z), thetransmitted zeros will not be affected by any CIR realization. Moreover,the added random channel zeros will not match any zero in

^(K) with probability one, such that the transmitted zero-pattern can beblind (noncoherent) identified at the receiver. To obtain noiserobustness, communication systems in accordance with many embodiments ofthe invention utilize a Binary MOCZ (BMOCZ) constellation, where eachsymbol defines the coefficients of a polynomial of degree K given by thezeros (roots)

α k ∈ k = { R 2 ⁢ m k - 1 ⁢ e j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ k - 1 K ❘ m k ∈ 𝔽 2 } ,

which are uniformly placed on a circle of radius R or R⁻¹, selected bythe message bits m_(k), see FIG. 3 for K=6. The filled blue circlesdenote one possible zero constellation of a BMOCZ signal. The basisangle, separating the conjugate-reciprocal zeros pairs, is given byθ_(K)=2π/K. This constellation set can be given by all normalizedHuffman sequences

_(η) ^(K)={x∈

^(K+1)|a(K, η)=x*

, x₀>0}, i.e., by all x∈

^(K+1) with positive first coefficient and “impulsive-like”autocorrelation

a=a(K,η)=

=(−η,0_(K),1,0_(K),−η) with η=η(R,K)=(R ^(K) +R ^(−K))⁻¹  (7)

for some R>1. The absolute value of (7) forms a trident with one mainpeak at the center, given by the energy ∥x∥₂ ²=1, and two equalside-peaks of η∈[0, 1/2), see FIGS. 4A and 4B. FIG. 4B shows in bluecircles the Huffman trident, given by the autocorrelation (7) of aHuffman sequence, and in red crosses the correlation of a Huffmansequence, given in FIG. 4A by blue circles, with the Huffman bracket oruniversal Huffman sequence, as pictured with red crossed in FIG. 4A. TheBMOCZ symbols can be robust against noise if

R _(uni) =R(K)=√{square root over (1+sin(π/K))}>1,K≥2.  (8)

Hence, the BMOCZ constellation set

_(*) ^(K)=

_(η*) ^(K) with η_(*)=η(R(K), K) is only determined by the number K.From the received N=L+K noisy signal samples (no CFO and no TO)

$\begin{matrix}{{y_{n} = {{{\sum\limits_{k = 0}^{K}{x_{k}h_{n - k}}} + w_{n}} = {\left( {x*h} \right)_{n} + w_{n}}}},\mspace{14mu}{n \in \lbrack N\rbrack}} & (9)\end{matrix}$

the BMOCZ decoder can perform Direct Zero-Testing (DiZeT) for eachreciprocal zero pair

on the received polynomial magnitude |Y(z)|=|Σ_(n=0) ^(N-1) y_(n)z^(n)|and can decide for the most likely one

$\begin{matrix}{{\overset{\hat{}}{m}}_{k} = \left\{ {{{\begin{matrix}{1,} & {{{{Y\left( {Re}^{j\; 2\pi\frac{k - 1}{K}} \right)}} < {R^{N - 1}{{Y\left( {R^{N - 1}e^{j\; 2\pi\frac{k - 1}{K}}} \right)}}}},} \\{0,} & {otherwise}\end{matrix}k} = 1},\ldots\mspace{14mu},K,} \right.} & (10)\end{matrix}$

where the inside zero samples are weighted accordingly.

Therefore, a global phase in Y(z) will have no effect on the decoder.But the CFO ϕ modulates the BMOCZ symbol in (4) and causes a rotation ofits zeros by −ϕ in (5), which complicates the hypothesis test of theDiZeT decoder. Hence, communication systems in accordance with manyembodiments of the invention estimate ϕ and/or use an outer code forBMOCZ to be invariant against an arbitrary rotation of the entirezero-set

^(K). In many embodiments, before decoding can be performed, the TO ofthe symbol which yields to the convolution (9) is identified. Due to thesymmetry of the zero-pattern and the energy concentration of the Huffmansequences, the BMOCZ design can address TO and CFO robustness andestimation, as is discussed further below.

Timing Offset and Effective Delay Spread for BMOCZ

In an asynchronous communication, the receiver typically does not knowwhen a packet from a transmitter (user) will arrive. Hence, incommunication systems in accordance with many embodiments of theinvention the receiver initially must detect a transmitted packet, whichis already one bit of information. It can be assumed that the receiverdetects the transmitted packet correctly and that in an observationwindow of N_(obs)=N_(noise)+K+L received samples, one single MOCZ packetof length K with maximal channel length of L is captured. By assuming amaximal length L and a known or a maximal K at the receiver, theobservation window can be chosen, for example, as N_(obs)=2N. From thenoise floor knowledge at the receiver, a simple energy detector with ahard threshold over the observation window can be used for a packetdetection. An unknown τ₀∈[N_(noise)] and CFO ϕ∈[0, 2π) can also bepresent in the observation window

{tilde over (r)} _(n) =e ^(jnϕ)(x*h)_(n-τ) ₀ +{tilde over (w)}_(n)=({tilde over (x)}*{tilde over (h)}T ^(τ) ⁰ )_(n) +{tilde over (w)}_(n) ,n∈[N _(obs)].  (11)

The challenge here is to identify r_(o) and the efficient channel lengthwhich contains most of the energy of the instantaneous CIR realizationh. In conventional communication systems, the estimation of thisTime-of-Arrival (TOA) parameter can be performed by observing the samechannel under many symbol transmissions, to obtain a sufficientstatistic of the channel PDP. As noted above, where only one or a smallnumber of observations are available, obtaining a good estimation can bevery challenging.

The efficient (instantaneous) channel length L_(ef), defined by anenergy concentration window, is likely to be much less than the maximalchannel length L, due to blockage and attenuation by the environment,which might also cause a sparse, clustered, and exponential decayingpower delay profile. For the MOCZ scheme, it can be beneficial tocorrectly identify in the window (11) the first received sample h₀x₀from the transmitted symbol x, or at least to not miss it, since it willcarry most of the energy if h₀ is the line of sight (LOS) path. For theoptimal radius in BMOCZ, x₀ typically carries on average 1/4 to 1/5 ofthe BMOCZ symbol energy, see also FIGS. 4A and 4B. An overestimatedchannel length L_(ef) can reduce overall bit-error performance, becausethe receiver can collect unnecessary noise samples.

Given an assumption of no CSI at the receiver, the channelcharacteristic, i.e., the instantaneous power delay profile, isdetermined entirely from the received MOCZ symbol. In a number ofembodiments that utilize a BMOCZ scheme, the radar properties of theHuffman sequences can be exploited to obtain estimates of the timingoffset and the effective channel delay in moderate and high SNR.

Huffman sequences have an impulsive autocorrelation (7), originallydesigned for radar applications, and are therefore very suitable tomeasure the channel impulse response. If the received signal r in (4)has no CFO and the transmitted sequence x is known at the receiver,correlation can be performed with x (matched-filter) and decision madebased upon sample maximal energy

$\begin{matrix}{\overset{\hat{}}{\tau} = {{H(r)} = {{\arg{\max\limits_{n \in {\lbrack{N_{obs} - K}\rbrack}}{\left( {r*\overset{\_}{x^{-}}} \right)_{n + K^{2}}}^{2}}} = {\arg{\max\limits_{n \in {\lbrack{N_{obs} - K}\rbrack}}{{\left( {{a*{hT}_{r}} + {w*\overset{\_}{x^{-}}}} \right)_{n + K}}^{2}.}}}}}} & (12)\end{matrix}$

The above approach can be referred to as the HufED (Huffman EnergyDetector), since the filtering with the conjugate-reciprocal sequencewill detect in an ideal AGWN channel the noise distorted Huffman energy.When the transmitted Huffman sequence is unknown at the receiver, thereceived signal cannot be correlated with the correct Huffman sequenceto retrieve the CIR. Instead, an approximative universal Huffmansequence can be used, which is just the first and last peak of a typicalHuffman sequence, expressed by the impulses (δ₀)_(k)=δ_(0k) and(δ_(K))_(n)=δ_(Kk) for k∈[K+1] as

_(ψ)=δ₀ +e ^(jψδ) ^(K) ∈

^(K+1)  (13)

which an be referred to as the K-Huffman bracket of phase ψ. Since thefirst and last coefficients are

x ₀=√{square root over (R ^(2∥m∥) ¹ ^(−K) ^(η) )},x _(K)=−√{square rootover (R ^(−2∥m∥) ¹ ^(+K) ^(η) )}  (14)

typical Huffman sequences, i.e., having same amount of ones and zeros,will have

|x ₀|² =|x _(K)|²=η.  (15)

By convolving the modulated Huffman sequence {tilde over (x)} with theHuffman bracket

_(ψ) the locational properties of the Huffman autocorrelation (trident)are maintained

a ˘ = ψ - _ * x ~ = ( e - j ⁢ ⁢ ψ ⁢ x 0 , e - j ⁢ ⁢ ψ ⁢ x ~ . , x 0 + e j ⁡ ( K⁢⁢ϕ - ψ ) ⁢ x K , x ~ . , e jK ⁢ ⁢ ϕ ⁢ x K ) , ⁢ ( 16 ) x ~ = x ~ ∘ + x ~ . ,

where x̊=x₀δ₀+x_(K)δ_(K) denotes the exterior signature and {dot over(x)}=(0, x₁, . . . , x_(K−1), 0) the interior signature of the Huffmansequence x, see FIG. 4A. Here, the interior signature can be seen as thedata noise floor distorting the trident a in (7). Taking theabsolute-squares in (16), the three peaks of the approximated tridentcan be obtained as follows

|{hacek over (a)} ₀|² =|x ₀|² ,|{hacek over (a)} _(K)|² =|x ₀|² +|x_(K)|²−2|x ₀ x _(K)|cos(Kϕ−ψ),|{hacek over (a)} _(2K)|² =|x_(K)|²,  (17)

where the side-peaks have energy

E _(sides) =|x ₀|² +|x _(K)|²=(R ^(2∥m∥) ² ^(−K) +R ^(−2∥m∥) ¹^(+K))η≤∥x∥ ²=1.  (18)

Since ϵ=2∥m∥₁−K∈[−K, K] we have 2≤R^(ϵ)+R^(−ϵ)≤R^(K)+R^(−K) and2η≤E_(sides)≤1, where the lower bounds are achieved for typicalsequences with ϵ=0 (having the same amount of ones and zeros) and theupper bounds for ϵ=±K (all ones or all zeros). If η=1/2 then the twocoefficients (the exterior signature x̊) will carry all the energy of theHuffman sequence. But then also R=1 and the only Huffman sequences (realvalued first and last coefficient) are given by x₀=±1/√{square root over(2)}, x_(K)=±1/√{square root over (2)} and x_(k)=0 for kc else, whichare the coefficients of polynomials with K uniform zeros on the unitcircle. For R given by (8) the autocorrelation side-lobe η isexponentially decaying in K but is bounded to η≥1/5 for K=1, 2, . . . ,512. Hence, E_(sides)≥0.4, such that almost half of the Huffman sequenceenergy is always carried in the two peaks.

If the CFO were known, ψ=Kϕ−π could be set to obtain the center peak in(17)

E _(center) =|x ₀|² +|x _(K)|²+2|x ₀ ∥x _(K)|≥4η≥0.8,  (19)

i.e., the energy of the center peak is roughly twice as large as theenergy of the side-peaks, and reveals the trident in the approximatedHuffman autocorrelation {hacek over (a)}. But, since the CFO is unknownin many embodiments of the invention and Kϕ−ψ≈2nπ for some n∈

then x₀+x_(K)≈0 for typical Huffman sequences, such that the power ofthe center peak will vanish. Hence, in the presence of an unknown CFO,the center peak does not always identify the trident. Therefore thepositive Huffman bracket

₀ can be correlated with the absolute-square value of x or in presenceof noise and channel with the absolute-square of the received signal{tilde over (r)}, which will result approximately in

d=

*|{tilde over (r)}| ² =

*|{tilde over (x)}*{tilde over (h)}T ^(τ) ⁰ |² +

*|{tilde over (w)}| ² ≈|{hacek over (a)}| ² *|{tilde over (h)}T ^(τ) ⁰|² +|{tilde over (w)}| ²∈

₊ ^(N) ^(obs) ^(+K)  (20)

where {tilde over (w)} and {tilde over (w)} are colored noise and

|{hacek over (a)}| ²=(|x ₀|²,|

² ,E _(sides),|

² ,|x _(K)|²)  (21)

denotes the noisy trident which collects three times the instant powerdelay profile |h²| of the shifted CIR. These three echos of the CIR canbe separated where K≥L. The approximation in (20) can be justified bythe isometry property of the Huffman convolution. Briefly, L<K, thegenerated (banded) L×N Toeplitz matrix T_(x)=Σ_(k=0) ^(K) x_(k)T^(k),for any Huffman sequence x∈

(K), is a stable linear time-invariant (LTI) system, since the energy ofthe output satisfies for any CIR realization h∈

^(L)

∥x*h∥ ² =∥hT _(x)∥² =tr(hT _(x) T _(x) *h*)=tr(h*A _(x) h)=∥x∥ ²tr(h*h)=∥x∥ ² ∥h∥ ².  (22)

Here. A_(x)=T_(x)T_(x)* is the L×L autocorrelation matrix of x, which isthe identity scaled by ∥x∥² if L≤K. Hence, each normalized Huffmansequence, generates an isometric operator T_(x) having the beststability among all discrete-time LTI systems.

In several embodiments, instead of or in addition to taking theabsolute-squares before correlating with the bracket

_(x), the receiver performs an average over different brackets. Inseveral embodiments, the receiver takes P≥2 uniform phases ψ_(p)=2πp/Pfor p∈[P]. The receiver can then consider the average energy of all Pbracket-filters, defining the filter

F P , n ⁡ ( x ~ ) = 1 P ⁢ ∑ p = 0 P - 1 ⁢  x ~ * ψ p  n 2 , n ∈ [ N ] . (23 )

For n=K

F P , K ⁡ ( x ~ ) = 1 P ⁢ ∑ p = 0 P - 1 ⁢  x ~ ⁢ * ψ p  K 2 =  x 0  2 + x K  2 - 2 P ⁢  x 0  ⁢  x K  ⁢ ∑ p = 0 P - 1 ⁢ cos ⁢ ⁢ ( K ⁢ ⁢ ϕ - 2 ⁢ π ⁢ ⁢p P ) . ( 24 )

Since

${\sum\limits_{p = 0}^{P - 1}{\cos\left( {{K\;\phi} - \frac{2\pi\; p}{P}} \right)}} = {{Re}\left( {e^{{jK}\;\phi}{\sum\limits_{p = 0}^{P - 1}e^{- j^{\frac{2\pi\; p}{P}}}}} \right)}$

for any Kϕ∈

the term will vanish for any 2≤P∈

by the orthogonality of the Fourier basis

${\frac{1}{P}{\sum\limits_{p = 0}^{P - 1}\;{{e^{{{- j}\frac{2\;\pi\; p}{P}} =}\left( {F_{P}\delta_{0}} \right)}^{H}F_{P}\delta_{1}}}} = {{\delta_{0}^{H}\delta_{1}} = 0.}$

To reduce the effort a selection of P=2 can be made to yield to thecenter energy |x₀|²+|x_(K)|²=E_(sides)≥2η for any normalized Huffmansequence. The other filter outputs are bounded by

${\max\limits_{n \neq K}\;{F_{2,n}\left( \overset{\sim}{x} \right)}} = {{\max\limits_{n \neq K}{\frac{1}{2}\left( {{{\overset{\sim}{x}*\kappa_{0}}}_{n}^{2} + {{\overset{\sim}{x}*\kappa_{\pi}}}_{n}^{2}} \right)}} = {\max\limits_{k \in {\lbrack{K + 1}\rbrack}}{{x_{k}}^{2}.}}}$

In fact, the filter F_(2,n) can be realized for any received signal y bytaking the first approach, i.e. by taking the absolute-square of thereceived signal and then filtering with the bracket

₀. i.e., F_(2,n)(y)=(|y|²*

₀)_(n). Note, that it holds trivially max{|x₀|²,|x_(K)|²}<F_(2,K)({tilde over (x)}) for K≥2, η∈[0, 1/2], ϕ∈[0, 2π) andany x∈

(η). Moreover, the following holds

$\begin{matrix}{{\max\limits_{k \in {\{{1,2,\ldots\mspace{14mu},{K - 1}}\}}}{x_{k}}^{2}} \leq {1 - \left( {{{x_{0}}^{2} + {x_{K}}^{2}} \leq {1 - {2\;{\eta.}}}} \right.}} & (26)\end{matrix}$

Processes that can be utilized to perform timing offset estimation inaccordance with various embodiments of the invention are discussedfurther below.

Timing Offset Estimation

The delay of the strongest path |h_(s)|=∥h∥_(∞) can be identified fromthe maximum in (20)

$\begin{matrix}{{\overset{\hat{}}{t} = {{\left( {\underset{t \in {\lbrack{N_{obs} + K}\rbrack}}{\arg\;\max}\; d_{t}} \right) - K} = {\left( {\underset{t \in {{\lbrack{N_{obs} - K}\rbrack} + K}}{\arg\;\max}\; d_{t}} \right) - K}}},} & (27)\end{matrix}$

where the last equality follows from the fact that both peaks

₀ are contributing between K and N_(obs). If the CIR has a LOS path,then s=0 and an estimate for the timing-offset can be found by{circumflex over (τ)}₀={circumflex over (t)}.

When the receiver takes an average over different brackets, simulationsreveal the existence of a typical sequence attaining an energy peak of1−2η, for even K see FIG. 5A. The maximal absolute-square value(peak-power) over η of all Huffman sequences is plotted by a bluediamond line. Typical sequences, denoted by dotted line, achieve themaximal peak-power, whereas atypical or odd-length Huffman sequenceshave a lower peak-power. Due to the energy normalization, such sequencescannot have more than 3 non-zero coefficients by (26). In fact, it canbe shown that for even K the center coefficient x_(K/2) carries theremaining energy. Opposed to codebooks with K odd, where typicalsequences have one more or one less zero outside than inside, the boundin (26) is not achievable. If the receiver searches for the energy ofthe trident the TO estimation of {tilde over (x)}_(τ)=xT_(τ)M_(ϕ) can beimproved by the average Trident Energy Detector (aveTriED), whichcollects the energy of all three peaks

$\begin{matrix}{\tau_{0} = {{\arg\;{\underset{n}{\;\max}{F_{2,{n + K}}\left( {\overset{\sim}{x}}_{r} \right)}}} + {2{F_{2,{n + {2K}}}\left( {\overset{\sim}{x}}_{r} \right)}} + {F_{2,{n + {3K}}}\left( {\overset{\sim}{x}}_{r} \right)}}} & (28)\end{matrix}$

for any K≥2, τ₀∈

, ϕ∈

, 1/2≥η>1/4 and x∈

_(η) ^(K). Here, the center peak is weighted by a factor of two, sinceit carries the energy of both side-peaks. To increase the maximal filteroutput further, a maximal energy filter can be utilized in accordancewith many embodiments of the invention, which is defined by

$\begin{matrix}{{{\overset{\sim}{F}}_{P,n}\left( \overset{\sim}{x} \right)} = {{\max\limits_{p \in {P}}{{\overset{\sim}{x}*\kappa_{2\;\pi\;{p/P}}}}_{n}^{2}} = \left\{ \begin{matrix}{{\max\limits_{p \in {P}}{{\overset{\sim}{x}*\kappa_{2\;\pi\;{p/P}}}}^{2}},} & {n = K} \\{{x_{n\mspace{14mu}{mod}\; K}}^{2},} & {{n\mspace{14mu}{else}},}\end{matrix} \right.}} & (29)\end{matrix}$

where for n=K the center energy is increased by maximizing over the Pphases

$\begin{matrix}{{{\overset{\sim}{F}}_{K}\left( \overset{\sim}{x} \right)} = {{\max\limits_{p \in {\lbrack P\rbrack}}{{\overset{\sim}{x}*\kappa_{2\;\pi\;{p/P}}}}_{K}^{2}} = {{x_{0}}^{2} + {x_{K}}^{2} - {2{x_{0}}{x_{K}}{\min\limits_{p \in {\lbrack P\rbrack}}{{\cos\left( {{2\;\pi\;{p/P}} - {K\;\phi}} \right)}.}}}}}} & (30)\end{matrix}$

Since Kϕ is uniformly distributed, induced by the channel CFO, p can beselected such that Kϕ−π∈[(p−1)/P, p/P]2π. Hence, the worst caseestimation (ϕ=pπ/KP for some p) would be−cos(π−π/P)=cos(π/P)≥1−2/P+(π−2)(P−2)/P²≥3/4 for P≥6, by the extendedKober's inequality, such that for any x∈

_(η) ^(K) and ϕ∈

it holds

$\begin{matrix}{{{\overset{\sim}{F}}_{6,K}\left( \overset{\sim}{x} \right)} = {{\underset{p \in {\lbrack 6\rbrack}}{\max\;}{{\overset{\sim}{x}*\kappa_{2\;\pi\;{p/6}}}}_{K}^{2}} > {{x_{0}}^{2} + {x_{K}}^{2} + {2{x_{0}}{x_{K}}\frac{3}{4}}} \geq {\frac{5}{2}{\eta.}}}} & (31)\end{matrix}$

Hence, for x∈

_(*) ^(K) and {tilde over (x)}_(τ)=xT_(τ) ₀ M_(ϕ) with τ∈

the maxTriED will identify the true TO

$\begin{matrix}{\tau_{0} = {\hat{n} = {{\underset{n}{argmax}{{\overset{\sim}{F}}_{6,{n + K}}\left( {\overset{\sim}{x}}_{\tau} \right)}} + {2{{\overset{\sim}{F}}_{6,{n + {2K}}}\left( {\overset{\sim}{x}}_{\tau} \right)}} + {{{\overset{\sim}{F}}_{6,{n + {3K}}}\left( {\overset{\sim}{x}}_{\tau} \right)}.}}}} & (32)\end{matrix}$

In FIG. 5B the results of simulations are illustrated for the failureprobability for various K's and fixed observation windowN_(obs)=4K_(max)=508 of the maxTriED and aveTriED over 5·10⁵ runs, bydrawing uniformly in each run a x∈

_(*) ^(K), a CFO ϕ∈[0, 2π), a TO τ∈[128], and i.i.d. additive noise w∈

(0, N₀)^(N). Furthermore, performance was simulated over ideal AWGNchannels and for multipath channels with L=10 paths, with LOS and powerdelay exponent p=0.95. As an ultimate comparison for an ED based TOAestimation, a delta pulse δ₀ was used as a transmit signal and thematched filter detector delED. In an ideal AWGN channel scenario at highSNR the same failure probability as delED is attained for maxTriED withP=6 and K_(max)=127 at only 4.5 dB more, see FIG. 5B. The performanceloss can be explained by the power distribution, since the energy on thetwo peaks is roughly η_(*)˜1/5. Furthermore, the maxTriED can onlyinfluence the CFO of one peak. If no CFO were present, and the Huffmansequence would be known at the receiver, the HufED detector in (12)would obtain the same performance as the delED detector. It should benoted that a more sophisticated estimation can be utilized, which candetect an error by a missing trident structure or by exploitingsoft-threshold and/or repetition/diversity methods. As can readily beappreciated, the specific detector utilized to detect. TO largelydepends upon the requirements of specific applications.

In case of NLOS or if the first paths are equally strong, then processesin accordance with several embodiments of the invention can go furtherback and identify the first significant peak above the noise floor,since the convolution sum of the CIR with the interior signature mightproduce a significant peak. It should be noted that this might result ina misidentification of the trident's center peak by (27), for example if|h_(s)|/|h₀|>>1. Therefore, processes in accordance with severalembodiments of the invention can utilize a peak threshold

$\begin{matrix}{{\rho = {{\rho\left( {K\; d} \right)} = {\frac{1}{K + 1}{\sum\limits_{n = {\hat{s} + {\hat{\tau}}_{0}}}^{\hat{s} + {\hat{\tau}}_{0} + K}\; d_{n}}}}},} & (33)\end{matrix}$

which is the average power of the Huffman sequence distorted by thechannel and noise. The noise-dependent threshold can be set based uponthe Noise Power N₀ to

ρ₀=max{ρ(K,d)/10, . . . ,ρ(K,d)/100,10·N ₀}  (34)

By using an iterative back stepping in Algorithm 1 as shown in FIG. 6,the process can stop if the sample power falls below the threshold ρ₀,which finally yields an estimate {circumflex over (τ)}₀ of thetiming-offset. In line 3 of Algorithm 1 in FIG. 6 the timing-offsetestimate is updated, if the sample power is larger than the threshold ρ₀and the average power of the preceding samples is larger than thethreshold divided by 1+(log b)/3, which will be weighted by the amount bof back-steps. We call the algorithm in FIG. 6 therefore acenter-peak-back-step (CPBS) algorithm. As can readily be appreciated,any of a variety of thresholds and/or algorithms can be utilized indetermining timing offsets as appropriate to the specific MOCZ schemeutilized and or the specific requirements of particular applications inaccordance with various embodiments of the invention. Processes forperforming channel length estimation in accordance with certainembodiments of the invention are discussed further below.

Efficient Channel Length Estimation

Since a communication system that utilizes BMOCZ does not, need anychannel knowledge at the receiver, it is also well suited for estimatingthe channel itself at the receiver. Here, a good channel lengthestimation is helpful for the performance of the decoder, particularlyif the power delay profile (PDP) is decaying. At some extent, thechannel delays will fade out exponentially and the receiver can cut-offthe received signal by using a threshold such as (but not limited to) acertain energy ratio threshold. The average received SNR is

$\begin{matrix}{{r\; S\; N\; R} = {\frac{{\mathbb{E}}\left\lbrack {{x*h}}^{2} \right\rbrack}{{\mathbb{E}}\left\lbrack {w}^{2} \right\rbrack} = {{E\frac{{\mathbb{E}}\left\lbrack {h}^{2} \right\rbrack}{N \cdot N_{0}}} = \frac{1}{N_{0}}}}} & (35)\end{matrix}$

where E=∥x∥²=N is the energy of the BMOCZ symbols, which is constant forthe codebook. If the power delay profile is flat, then the collectedenergy is uniform and the SNR is unlikely to change if the channellength at the receiver is cut. However, the additional channel zeros canincrease the confusion for the DiZeT decoder and may reduce BERperformance. Therefore, the performance can decrease for increasing L ata fixed symbol length K. For the most interesting scenario of K≈L theBER performance loss is only 3 dB over E_(b)/N₀, but will increasedramatically if L>>K. The reason for this behaviour is the collection ofmany noise taps, which can lead to more distortion of the transmittedzeros. Since in most realistic scenarios the PDP will be decaying, mostof the channel energy is typically concentrated in the first channeltaps. Hence, cutting the received signal length to N_(eff)=K+L_(eff),can reduce the channel length to L_(eff)<L and improve the rSNR fornon-flat PDPs with p<1, since it holds

$\begin{matrix}{\frac{1}{N_{0}} = {{\frac{N\left( {{\sum\limits_{l = 0}^{L_{{eff} - 1}}\; p^{l}} + {p^{L_{eff}}{\sum\limits_{l = 0}^{L - L_{eff}}p^{l}}}} \right)}{N \cdot N_{0}} \approx \frac{N{\sum\limits_{l}^{L_{eff} - 1}\; p^{l}}}{\left( {K + L} \right)N_{0}} < \frac{N{\sum\limits_{l}^{L_{eff} - 1}\; p^{l}}}{\left( {K + L_{eff}} \right)N_{0}}} = {\frac{N\;{{\mathbb{E}}\;\left\lbrack {h_{eff}}^{2} \right\rbrack}}{N_{eff}N_{0}}.}}} & (36)\end{matrix}$

Since 1/

[∥h_(eff)∥²]<<N/N_(eff) a significant gain in SNR can be obtained ifL>>K and p<1. Hence, by cutting the received signal to the effectivechannel length, given by a certain energy concentration, the SNR, can beimproved and, at the same time, the amount of channel zeros can bereduced, which is demonstrated in the simulations discussed below.

Assuming knowledge of the noise floor N₀ at the receiver, a cut-off timecan be defined as a window of time which, for example, contains 95%, ofthe received energy. As can readily be appreciated, the cut-off time canbe determined using any of a variety of techniques appropriate to therequirements of a specific application in accordance with variousembodiments of the invention.

In several embodiments of the invention, the estimation of the channellength L_(eff) is performed after the detection of the timing-offset τ₀with Algorithm 1 from FIG. 6. It can be assumed that the maximal channeldelay is L. Since the BMOCZ symbol length is K+1, it is known that thesamples r_(h)=(r_(τ) ₀ _(+K+1), . . . , r_(τ) ₀ _(+K+L)) of the receivedtime-discrete signal in (4), which is the CIR, correlated by shifts of xand distorted by additive noise (ignoring here the CFO distortion sinceit is not relevant for the PDP estimation), see FIGS. 7A and 7B. FIG. 7Bshows the received signal in blue, given by the Huffman sequence in bluecircles and the CIR in red crosses, pictured in FIG. 7A. The lastchannel tap h_(L) _(eff) will be multiplied by |x_(K)|², which can be asstrong as |x₀|² on average.

There are many signal processing methods that can be utilized to detectthe efficient energy window N_(eff) in the received samples, including(but not limited to) total variation smoothing, or regularizedleast-square methods which promotes short window sizes (sparsity). Inseveral embodiments, the process (i.e. Algorithm 2) shown in FIG. 8 isutilized, which iteratively increases L_(eff) starting at 1 andincreasing until enough channel energy is collected. The estimatechannel/signal energy can be set to

E _(r) =∥r _(h)∥² −LN ₀  (37)

starting with the maximal CIR length L. By assuming a path exponent ofp, processes in accordance with various embodiments of the invention cancalculate a threshold for the effective energy E_(eff)≃μE_(r) withμ=p^(LN) ⁰ ^(/2E) ^(r) . The process can then collect as many samplesL_(eff) of r_(h) until the energy E_(eff) is achieved and setN_(eff)=K+L_(eff). The extracted modulated signal can then be given by

{tilde over (y)}=({tilde over (r)} _({circumflex over (τ)}) ₀ ,{tildeover (r)} _({circumflex over (τ)}) ₀ ₊₁ . . . ,{tilde over (r)} _(τ) ₀_(+N) _(eff) ⁻¹),  (38)

which can be further processed for CFO detection and final decoding asis discussed further below.

While specific processes are described above for performing channellength estimation, any of a variety of techniques for performing channellength estimation can be utilized as appropriate to the requirements ofspecific applications in accordance with various embodiments of theinvention. Processes for determining CFOs in accordance with variousembodiments of the invention are discussed further below.

Carrier Frequency Offset

When processes similar to those described above are utilized to estimatetiming offset, processes in accordance with several embodiments of theinvention can attempt to estimate a CFO assuming that the down-convertedbaseband signal in (11) has no further timing-offset and captured allpath delays up to N=K+L. Under these assumptions, the signal can furtherbe assumed to experience an unknown CFO of ϕ∈[0, 2π)

{tilde over (y)} _(n) =e ^(jnϕ)(x*h)_(n) +w _(n) ,n∈[N].  (39)

This is a common problem in many multi-carrier systems, such as OFDM,which therefore often require CFO estimation algorithms as noted above.For a bandwidth of W=1/T, the relative frequency offset is

ϵ=Δf·T=Δf/W.  (40)

Given a carrier frequency of f_(c)=1 GHz with a drastic frequency offsetof Δf∈[−1, 1] MHz and bandwidth W=1 Mhz, this would result in a relativefrequency offset ϵ∈[−1, 1], which is able to rotate all zeros by anyϕ=2πϵ∈[0, 2π) in the z-plane. Hence, the received polynomial (noiseless)will experience a rotation of all its N−1 zeros by the angle ϕ

$\begin{matrix}{{\overset{\sim}{Y}(z)} = {{\sum\limits_{n = 0}^{N - 1}\;{{\overset{\sim}{y}}_{n}z^{n}}} = {{\sum\limits_{n = 0}^{N - 1}\;{y_{n}e^{j\; n\;\phi}z^{n}}} = {{Y\left( {e^{j\;\phi}z} \right)} = {{y_{N - 1}{\prod\limits_{n = 1}^{N - 1}\;\left( {{e^{j\;\phi}z} - \gamma_{n}} \right)}} = {{e^{j\;{\phi{({N - 1})}}}y_{N - 1}{\prod\limits_{n = 1}^{N - 1}\;\left( {z - {\gamma_{n}e^{{- j}\;\phi}}} \right)}} = {e^{{- j}\;{\phi{({N - 1})}}}x_{K}h_{L - 1}{\prod\limits_{k = 1}^{K}\;{\left( {z - {\alpha_{k}e^{{- j}\;\phi}}} \right){\prod\limits_{l = 1}^{L - 1}\;{\left( {z - {\beta_{l}e^{{- j}\;\phi}}} \right).}}}}}}}}}}} & (40)\end{matrix}$

As illustrated in FIG. 3 for K=6, each rotated zero (red) {tilde over(α)}_(k)=e^(−jϕ)α_(k) ideally does not leave the zero-codebook set(blue)

$= {{(K)}:={\left\{ {{R^{\pm 1}e^{j\; 2\;\pi\frac{k}{K}}}❘{k \in \lbrack K\rbrack}} \right\}.}}$

To apply the DiZeT decoder, decoding processes in accordance with manyembodiments of the invention find θ such that e^(jθ)α∈

, i.e., ensuring that all the K data zeros will lie on the uniform grid.Hence, for θ_(K)=2π/K the CFO can be split into

ϕ=lθ _(K)+θ  (42)

for some l∈[K] and θ∈[0, θ_(K)), where l is called the integer and θ thefractional CFO. When θ=0 (or is correctly compensated), the DiZeTdecoder can sample at correct zero positions and decode a cyclicpermuted bit sequence {tilde over (m)}=mS^(l) of the transmittedmessage. The cyclic permutation is due to the unknown integer shift l,which can be corrected by use of a cyclically permutable code to encodethe transmitted data in accordance with various embodiments of theinvention and as is discussed further below.

Decoding BMOCZ via FFT

In many embodiments, the DiZeT decoder for BMOCZ can allow a simplehardware implementation at the receiver. In a number of embodiments, thereceived samples y_(n) are scaled with the radius powers R^(n)respectively R^(−n)

$\begin{matrix}{{y\; D_{\underset{\_}{R}}}:={{y\begin{pmatrix}1 & 0 & \ldots & 0 \\0 & R & \ldots & 0 \\\vdots & \; & \ddots & \vdots \\0 & 0 & \ldots & R^{N - 1}\end{pmatrix}}.}} & (43)\end{matrix}$

By applying the Ñ-point unitary I)FT matrix F* on the N₀ zero-paddedscaled signal, where Ñ=QK with Q:=┌N/K┐, the samples of the z-transformcan be obtained

${\sqrt{N}{y\left( {D_{\underset{\_}{R}}0_{N,{{Q\; K} - N}}} \right)}F^{*}} = {\left( {{\sum\limits_{n = 0}^{N - 1}{y_{n}R^{n}e^{j\; 2\pi\frac{0 \cdot n}{N}}}},\ldots\mspace{14mu},{\sum\limits_{n = 0}^{N - 1}{y_{n}R^{n}e^{j\; 2\pi\frac{{({\overset{\sim}{N} - 1})} \cdot n}{N}}}}} \right) = {Y\left( \alpha_{Q}^{(1)} \right)}}$

where

α Q , k ( m ) = α k ( m ) ⁡ ( e 0 , … ⁢ , e j ⁢ ⁢ 2 ⁢ ⁢ π ⁢ Q - 1 QK ) ∈ Q .

Hence, the DiZeT decoder simplifies to

$\begin{matrix}{{\overset{\hat{}}{m}}_{k} = \left\{ {{{\begin{matrix}{1,} & {{{{y{\overset{\sim}{D}}_{\underset{\_}{R}}F^{*}}}_{Q{({k - 1})}} < {R^{K - 1}{{y{\overset{\sim}{D}}_{\underset{\_}{R} - 1}F^{*}}}_{Q{({k - 1})}}}},} \\{0,} & {else}\end{matrix}k} = 1},\ldots\mspace{14mu},{K.}} \right.} & (44)\end{matrix}$

Here, Q≥2 can be seen as an oversampling factor of the IDFT, where eachQth sample point is picked to obtain the zero sample values. Hence, inmany embodiments the decoder can be fully implemented by a simple IDFTfrom the delayed amplified received signal, by using for example FPGA oreven analog front-ends. The diagonal scaling matrix (43) can berewritten in the symmetric form

$\begin{matrix}{{D_{R}:={{{diag}\left( {R^{{({N - 1})}/2},\ldots\mspace{14mu},R^{{- {({N - 1})}}/2}} \right)} = {R^{{- {({N - 1})}}/2}D_{\underset{\_}{R}}}}},} & (45)\end{matrix}$

such that D_(R) ⁻¹=D_(R) ⁻¹ corresponds to a time-reversal of thediagonal, which results in

|t{tilde over (D)} _(R) F* _(QK)|_(Qk)≤|

{tilde over (D)} _(R) F* _(QK)|_(Qk) ,k∈[K],  (46)

since the absolute values cancel the phases from a circular shift S andthe conjugate-time-reversal

=ySΓ, where Γ=F² is the circular time-reversal, can be rewritten byusing F*Γ=F=F*.

Fractional CFO Estimation Via Ovcrsampled FFTs

To estimate the factional frequency offset, oversampling can beperformed in many embodiments of the invention by choosing Q>┌N/K┐ toadd Q further K zero blocks to D_(R). This can lead to an oversamplingfactor of Q and allows quantization of [0, θ_(K)) in Q uniform bins withseparation ϕ_(Q)=θ_(K)/Q for a base angle θ_(K)=2π/K. Hence, theabsolute values of the sampled z-transform in (39) of the rotatedcodebook-zeros are given by

$\begin{matrix}{{{{\overset{\sim}{Y}\left( {e^{{jq}\;\phi_{Q}}\alpha_{k}^{(1)}} \right)}} = {{\overset{\sim}{y}\;{\overset{\sim}{D}}_{R}F_{QK}^{*}}}_{{Q\; k} \oplus_{\overset{\sim}{N}}q}},} & (47) \\{{{\overset{\sim}{Y}\left( {e^{j\; q\;\phi_{Q}}\alpha_{k}^{(0)}} \right)}} = {{\overset{\_}{{\overset{\sim}{y}}^{-}}{\overset{\sim}{\; D}}_{R}F_{QK}^{*}}}_{{Qk} \oplus_{\overset{\sim}{N}}q}} & \;\end{matrix}$

for each q∈[Q] and k∈[K], where ⊕_(N) is addition modulo N. To estimatethe fractional frequency offset of the base angle, the K smaller samplevalues can be summed and the fraction corresponding to the smallest sumselected

$\begin{matrix}\hat{\begin{matrix}\begin{matrix}{{\overset{\hat{}}{q} = {\arg{\min\limits_{q\; \in {\lbrack Q\rbrack}}{\sum\limits_{k = 1}^{K}{\min\left\{ {{{{\overset{\sim}{y}}_{m}{\overset{\sim}{D}}_{R}F_{QK}}}_{{Qk} \oplus_{\overset{\sim}{N}}q},{{\overset{\_}{{\overset{\sim}{y}}_{m}^{-}}{\overset{\sim}{D}}_{R}F_{QK}}}_{{Qk} \oplus_{\overset{\sim}{N}}q}} \right\}}}}}}{{{and}{\;\mspace{11mu}}\hat{\theta}} = {\frac{\hat{q}}{Q}{\theta_{K}.}}}} & (73)\end{matrix} & (48)\end{matrix}} & (48)\end{matrix}$

Then the recovered signal ŷ={tilde over (y)}M^(−{circumflex over (θ)})will have the data zeros on the constellation grid

_(K). See FIG. 9 for the BER performance degradation of BMOCZ under arandom fractional CFO ϕ∈[0, θ₃₂] and FIG. 3 for a schematic picture ofthe fractional zero rotation.

Using Cyclically Permutable Codes

To be robust against rotations which are integer multiples of the baseangle, an outer block code in

₂ ^(K) can be used for the binary message m∈

₂ ^(B), which is invariant against cyclic shifts, e.g., a bijectivemapping on the Galois Field

₂={0, 1}

:

₂ ^(B)→

₂ ^(K) ,m

c=

(m)  (49)

such that

⁻¹(cS^(l))=m for any l∈[K]. The common notation for the code length n=Kcan be used. Such a block code is called a cycling register code (CRC)

_(CRC), which can be constructed from the linear block code

₂ ^(n), by separating it in all its cyclic equivalence classes

[c]_(CRC) ={cS ^(l) |l∈[n]},c∈

₂ ^(n)  (50)

where c has cyclic order v if cS^(v)=c for the smallest possible v∈{1,2, . . . , n}. To make coding one-to-one, each equivalence class can berepresented by the codeword {tilde over (c)} with smallest decimalvalue, also called a necklace. Then

₂ ^(n) is given by the union of all its M_(CRC) equivalence classrepresentatives and its cyclic shifts. i.e.

2 n = ⋃ i = 1 M CRC ⁢ { c ~ i ⁢ S l ⁢  ⁢ l = 1 , … ⁢ , v ⁡ ( i ) } ( 51 )

In a number of embodiments, the above expression can be utilized togenerate in a systematic way a look-up table for the cycling registercode. Unfortunately, the construction is non-linear and combinatorialdifficult. However, the cardinality of such a code can be provenexplicitly for any positive integer n to be (number of cycles in a purecycling register)

 CRC ⁢ ( n )  = 1 n ⁢ ∑ d ⁢  n ⁢ Φ ⁡ ( d ) ⁢ 2 n / d ( 52 )

where Φ(d) is the Euler function, which counts the number of elementst∈[d] coprime to d.

For n prime, the following can be obtained

 CRC ⁢ ( n )  = 1 n ⁢ ( 2 n + ( n - 1 ) ⁢ 2 ) ≥ 2 n n = 2 n - log 2 ⁢ n (53 )

which would allow a transmitter to encode at least B=n−┌log n┐ bits. Forn=K=31 this would result in a loss of only 5 bits and is similar to theloss in a BCH-(31,26) code, which can correct 1 bit error. Note, thecardinality of a cycling register code

_(CRC) is minimal if n is prime. This can be seen by acknowledging thefact that the cyclic order of a codeword is always a divisor of n.Therefore, if n is prime the trivial orders 1 and n are present, wherethe only codewords with order v=1 are the all one 1 and all zero 0codeword and all other codewords have the maximal or full cyclic orderv=n. Hence, extracting the CRC from

₂ ^(n) obtains the same cardinality (53). Taking from a block code oflength n only the codewords of maximal cyclic order n and selecting onlyone representative of them defines a cyclically permutable code (CPC).If n is prime, only two equivalence classes in

_(CR) are not of maximal cyclic order, hence the cardinality for a CPCif n is prime is at most (2^(n)−2)/n.

However, the construction of CPCs, even for n prime, is a combinatorialproblem, especially the decoding. Hence, to reduce the combinatorialcomplexity, many approaches starting from cyclic codes and extract allcodewords with maximal cyclic order. Since a cyclic (n, k, d_(min)) codecorrects up to (d_(min)−1)/2 bit errors, any CPC code extraction willinherit the error correction capability.

In many embodiments, a CPC can be constructed from a binary cyclic (n,k, d_(min)) code by still obtaining the best possible cardinality(2^(k)−2)/n if n is a Mersenne prime. An affine subcode can be extractedfrom the CPC with maximal dimension, which can be referred to as anaffine cyclically permutable code (ACPC). This allows a linear encodingof B=k−log(n+1) bits by a generator matrix and an additive non-zero rowvector, which defines the affine translation.

CPC Construction from Cyclic Codes

Cyclic codes exploit efficiently the algebraic structure of Galoisfields

_(q), given by a finite set having a prime power cardinality q=p^(m′). Alinear block code over

_(q) is a cyclic code if each cyclic shift of a codeword is a codeword.It is a simple-root cyclic code if the characteristic p of the field

_(q) is not a divisor of the block length n. If the block length is ofthe form n=q^(m)−1, it can be referred to as a primitive block lengthand if the code is cyclic it can be called a primitive cyclic code. In anumber of embodiments, binary cyclic codes of prime length n=2^(m)−1with q=p=2 are utilized, which are simple-root and primitive cycliccodes. Due to the linearity of cyclic codes, they can be encoded anddecoded by a generator G and check matrix H in a systematic way. Thecardinality of a binary cyclic (n, k) code is always M_(c)=2^(k). Hence,for cyclic codes of prime length n, the partitioning in equivalenceclasses of maximal cyclic order and selecting one codeword as theirrepresentative provides, leaves us with a maximal cardinality of

 CPC  ≤ 2 k - 1 n . ( 54 )

for any extracted CPC. Note, the zero codeword is always a codeword in alinear code but has cyclic order one and hence is typically not anelement of a CPC. To exploit the cardinality most efficiently, the goalis to find cyclic codes such that each non-zero codeword has maximalcyclic order. In several embodiments, codes are constructed for primecode lengths of the form n=2^(m)−1, also known as Mersenne primes. Form=2, 3, 5, 7 this applies to K=n=3, 7, 31, 127, which are relevantsignal lengths for binary short-messages. Furthermore, severalembodiments only consider cyclic codes which have 1 as a codeword. Sincen is prime, each codeword, except 0 and 1 has maximal cyclic order.Hence, each codeword of a cyclic (n, k) code has maximal cyclic orderand since it is a cyclic code all its cyclic shifts must be alsocodewords. Hence the cardinality of codewords having maximal order isexactly M=2^(k)−2. In a number of embodiments, the cyclic code ispartitioned in its cyclic equivalence classes, which leaves |

_(CPC)|=M/n=(2^(k)−2)/n. Note, this number is indeed an integer, by theprevious mentioned properties. The main advantages of the constructiondescribed herein (also referred to as the Kuribayashi-Tanaka (KT)construction) is the systematic code construction and the inheriterror-correcting capability of the underlying cyclic code, from whichthe CPC is constructed. Furthermore, combining error-correction andcyclic-shift corrections can provide advantages in many systems thatutilize BMOCZ in accordance with various embodiments of the invention.

In a number of embodiments, an inner

_(in)−(k, k−m) cyclic code is utilized in combination with an outer

_(out)−(n, k) cyclic code, where the inner cyclic codewords are affinetranslated by the Euclidean vector e₁=(1, 0, . . . , 0)∈

₂ ^(k). In this sense, the CPC construction is offline. This can berealized by an offline mapping from

₂ ^(k-m) to

₂ ^(n), which can be represented by the BCH generator matrices G_(in)and G_(out) together with the affine translation e₁ as

:

₂ ^(k-m)→

_(in) +e ₁→

_(out)⊂

₂ ^(n)

m

i=mG _(in) +e ₁

c=iG _(out)=

(m).  (55)

To derive the generator matrix the algebraic structure of the cycliccodes can be exploited, given by its Galois fields. By definition ofcyclic codes, the following polynomial is factorized

$\begin{matrix}{{x^{n} - 1} = {\left( {x - 1} \right){\prod\limits_{s = 1}^{S}{G_{s}(x)}}}} & (56)\end{matrix}$

in irreducible polynomials G_(s)(x) of degree m_(s), which are divisorsof m. If n=2^(m)−1 is prime, m=log(n+1) is also prime and hence all theirreducible polynomials are primitive and of degree m_(s)=m, except oneof them, G₀(x)=x−1, has degree one. Hence, it holds that S=(n−1)/m. Asouter generator polynomial

${G_{out}(x)} = {{\prod\limits_{s = {S - J + 1}}^{S}{G_{s}(x)}} = {\sum\limits_{i = 0}^{Jm}{g_{{out},i}x^{k}}}}$

several embodiments of the invention choose the product of the last1≤J≤S primitive polynomials G_(s)(x) yielding a degree Jm=n−k. Eachcodeword polynomial of degree less than n is then given by

C(x)=I(x)G _(out)(x)  (57)

where I(x)=Σ_(a=0) ^(k-1) i_(a)x^(a) is the informational polynomial ofdegree less than k and represented by the binary information word i=(i₀,i₁, . . . , i_(k−1))∈

₂ ^(k), which can be referred to as the inner codeword. Similar, thecodeword polynomial C(x) is represented by the CPC codeword c∈

₂ ^(n). A message polynomial M(x)=Σ_(b=0) ^(k-m-1) m_(b)x^(b) of degreeless than k−m and R(x)=1 will be mapped to the information polynomial

I(x)=M(x)G _(in)(x)+1,  (58)

by the inner generator polynomial G_(in)(x)=G₁(x), which has degree m.

All possible cyclic equivalence classes can be mapped to M(x). For J=1,all the possible S−1 primitive polynomials G_(s)(x) generate S−1distinct inner codes. However, the remaining S−2 inner codes will onlymap <2^(k-m) more message polynomials to codeword polynomials andtherefore are not enough to encode an additional bit. Hence, these otherinner codewords can be omitted. This has the advantage, that (55) can bewritten with the subset of the CPC as an affine cyclically permutablecode (ACPC), which is given by the polynomial multiplication over

₂

C(x)=I(x)G _(out)(x)=(M(x)G _(in)(x)+1)G _(out)(x)=M(x)G(x)+G_(out)(x).  (59)

Here, a third generator polynomial G(x)=G_(in)(x)G_(out)(x) can beintroduced which will be affine translated by the polynomial G_(out)(x).This generator polynomial G(x) will map surjective

₂ ^(k-m) to a cyclic code

₂ ^(n) and can therefore be expressed in matrix form as

mG+g _(out)∈

_(ACPC)⊂

_(out)  (60)

where g_(out)=(g_(out,0), . . . , g_(out,n-k), 0, . . . , 0)∈

₂ ^(n) and the generator matrix (Toeplitz) is given by k−m shifts of thegenerator polynomial coefficients g₀, . . . , g_(n-(k-m)) as

G = ( g 0 g 1 g k - m - 1 … g n - ( k - m ) 0 … 0 0 g 0 g k - m - 2 … gn - ( k - m ) - 1 g n - ( k - m ) … 0 ❘ ⋮ \ ⋮ 0 0 g 0 … … g n - ( k - m) ) ∈ 2 k - m × n . ( 61 )

For linear codes, the Euclidean algorithm can be used to computex^(n-i)=Q_(i)(x)G(x)+S_(i)(x) for i=1, . . . , k−m, where the remainderpolynomials

$\begin{matrix}{{S_{i}(x)} = {\sum_{j}{s_{i,j}x^{j}}}} & \;\end{matrix}$

will have degree less than n−(k−m), which allows one to rewrite theToeplitz matrices as systematic matrices. Here, the check symbolss_(i,j) define the k−m×n systematic generator and n−k+n×n check matrix

$\begin{matrix}{{{\overset{\sim}{G} = \left\lbrack {- {PI}_{k - m}} \right\rbrack},{\overset{\sim}{H} = {\left\lbrack {I_{n - k + m}P^{T}} \right\rbrack\mspace{14mu}{with}}}}{P = \begin{pmatrix}s_{{k - m},0} & \ldots & s_{{k - m},{n - k + m - 1}} \\s_{{k - m - 1},0} & \ldots & s_{{k - m - 1},{n - k + m - 1}} \\\vdots & \; & \vdots \\s_{1,0} & \ldots & s_{1,{n - k + m - 1}}\end{pmatrix}}} & (62)\end{matrix}$

such that {tilde over (G)}{tilde over (H)}^(T)=O_(k−m,n−k+m). Thisallows one to write the cyclic outer code aa

₂ ^(k-m) G=

₂ ^(k-m) {tilde over (G)}. Of course, each mG and m{tilde over (G)} canbe mapped to different codewords, but this is just a relabeling. Since

₂ ^(k-m){tilde over (G)} defines a cyclic (n, k−m) code and n is prime,each codeword has n distinct cyclic shifts. The affine translationg_(out) separates each of these n distinct cyclic shifts by mapping themto representatives of distinct cyclically equivalence classes of maximalorder. This can provide a very simple encoding rule for each m∈

₂ ^(B):

c=m{tilde over (G)}+g _(out)=

(m)∈

_(ACPC),  (63)

and decoding rule. Here an ACPC codeword c can be decoded by justsubtracting the affine translation g_(out) and cut-off the last B=k−mbinary letters to obtain the message word m∈

₂ ^(B), see (62). However, it can be observed that a cyclic shiftedcodeword v=cS^(l) and by construction that only one cyclic shift will bean element of

_(ACPC) and consequently it holds

∀j≠l mod(n−1):{tilde over (c)} _(j) =vS ^(−j) −g _(out)∉

₂ ^(k-m) {tilde over (G)}

{tilde over (c)} _(j) {tilde over (H)} ^(T)≠0.  (64)

Hence, it is only necessary to check all n cyclic shifts of thesense-word v to identify the correct cyclic shift l, which is given if{tilde over (c)}_(l){tilde over (H)}^(T)=0. If there is an additiveerror e, process in accordance with various embodiments of the inventioncan use the error correcting property of the outer cyclic code

_(out) in (59) to repair the codeword. The systematic matrices can beused {tilde over (G)}_(out) and {tilde over (H)}_(out) to represent theouter cyclic code in a systematic way. Note, that each informationmessage i which corresponds to a CPC codeword c will be an element of

₂ ^(k){tilde over (G)}_(out). Furthermore, all its cyclic shifts cS^(l)will be outer cyclic codewords. Hence, by observing the sense-word

{tilde over (v)}=cS ^(l) +e  (65)

and determining its syndrome s=v{tilde over (H)}_(out) ^(T), which islooked-up in the syndrome table T_(synd) of {tilde over (H)}_(out) toidentify the corresponding coset leader (error-word) e, which recoversthe shifted codeword (assuming maximal t errors, bounded-distancedecoder) as

v={tilde over (v)}−e,  (66)

This can allow the receiver to correct up to t=(n−k−1)/2 errors of thecyclic codeword. From the additive error-free sense-word c the processcan, as described previously, identify the correct shift {circumflexover (l)} from (64) and by taking the last

=k letters the original message m. If the error vector e introduces morethan t bit flips, the error correction is likely to fail and the chanceis high that the receiver will mix up the ACPC codewords and experiencea block (word) error. In many embodiments, a low coding rate is utilizedfor the outer cyclic code G_(out) to reduce the likelihood of such acatastrophic error. In several embodiments, the rate of the outer codecan adapt in response to measured channel conditions. In a number ofembodiments, the receiver can transmit a message containing at least oneof a selected code rate and/or channel measurement to coordinatechanging the rate of the outer code in the transmitter.

The identified cyclic shift and fractional CFO estimation (48) yieldsthen the estimated CFO

{circumflex over (ϕ)}={circumflex over (θ)}+{circumflex over (l)}θ_(K).  (67)

While specific processes are described above for estimating TO, and/orCFO in addition to correcting bit errors, any of a variety of codesincluding CRC, CPC, ACPC, and/or any other codes that can be utilized todetect cyclic permutations and/or receiver implementations can beutilized as appropriate to the requirements of specific applications inaccordance with various embodiments of the invention. Simulations ofvarious communication systems implemented using BMOCZ schemes andemploying cyclic codes to encode message bits in accordance with anumber of embodiments of the invention are discussed further below.

Implementation in Matlab

Simulations of communication systems employing MOCZ and cyclic codes canbe performed in Matlab by using the function primpoly(m, ‘all’) to listall primitive polynomials for

₂ _(m) . The first in the list can be selected for G_(in) and theproduct of the last l for G_(out). The Euclidean algorithm isimplemented by gfdeconv. Note, the decimal value of the binary words canbe given by m=m (2)=Σ_(b) m_(b)2^(b) and implemented by de2bi(vm). Thegenerator and check matrices can be constructed as well as the syndrometable T_(synd) for the error correction.

FIG. 16 shows the Matlab program (Code) for creating the systematicgenerator matrix G in line 20, defined by {tilde over (G)} in (62), andthe systematic check matrix tH in line 21, defined by {tilde over (H)}in (62), as well as the affine shift a in line 22, defined by g_(out)after (60), to encode binary messages m of length B=k−m to ACPCcodewords c of length n=2^(m)−1 by (63). Furthermore, a syndrome tableTsynd for the outer cyclic (BCH) code is created in line 30 and thecheck matrix tHout to correct by the syndrome additive errors in thesense-word {tilde over (v)}₁ as in (66). Note, that the value of k canbe chosen from a list, of BCH-(n,k) codes to correct up to t=(n−k−1)/2errors.

Decoding: First an additive error-correction is performed in the outercode

_(out), which works as for cyclic codes via the syndrome-table T_(synd)and the check matrix {tilde over (H)}_(out). First, the syndromes=v{tilde over (H)}_(out) ^(T) is computed and looked-up in thesyndrome-table to identify the corresponding coset leader (error-word)e, which yields the recovered shifted codeword (assuming maximal terrors, bounded-distance decoder) as

{tilde over (c)}=v−e.  (68)

From the additive error-free but circular shifted codeword {tilde over(c)} the process can construct all n shifts, subtract the constant shiftvector a=g_(out), and apply the check matrix {tilde over (H)}_(out) tofind the one shift which is most likely the ACPC codeword

Example 1. The most non-trivial example of ACPC is for m=3 and l=1,which gives

n=2³−1=7,k=7−3=4,B=4−1=1  (69)

This is also the Hamming (7, 4) code with minimal distance d_(min)=3 andhence can correct 1 bit error. The code is also perfect.

Example 2. For K=n=31 it is possible to get for t=2 error corrections amessage length of k=21 in a cyclic BCH-(31, 21) code from which it ispossible to construct a CPC of cardinality

 CPC  = ∑ i = 1 4 ⁢ 2 5 ⁢ ( i - 1 ) + 1 = 2 1 + 2 6 + 2 11 + 2 16 = 6750≥ 2 16 . ( 70 )

This allows communication systems in accordance with several embodimentsof the invention to encode B=16=k−mbits. The cardinality can be optimalfor any cyclic (31,26) code (54)

 CPC  = M c m = 2 21 - 2 31 = 2 ⁢ 2 20 - 1 2 5 - 1 = ∑ i = 0 3 ⁢ 2 5 ⁢i + 1 = 2 1 + 2 6 + 2 11 + 2 16 = 6750. ( 71 )

The next example would be for n=127, which is also simulated in FIGS.10A and 10B. Unfortunately, the next. Mersenne prime is only at 8191,which may not constitute a short-packet length in many applications.However, other CPC constructions can be utilized in certain embodimentsof the invention, which do not require Mersenne prime lengths n or evenbinary alphabets. For example only prime lengths can be required or/andnon-binary alphabet extensions which allow the usage of M-ary MOCZschemes. In many embodiments, CPCs are utilized with alphabet size q,given as a power of a prime, and block length n=q^(m)−1 for any positiveinteger m. Hence, for q=2 and a binary cyclic code (n=2^(m)−1, k,d_(min)) the code can result in (2^(k)−1)/n CPC codewords. An examplecan be given for n=15=2⁴−1 and k=8, d_(min)=4 where in is not a.Mersenne prime number and the maximal bound (54) of (2⁸−1)/15=17 isachieved, allowing a transmitter to encode B=4 bits and a receiver tocorrect 2 bit errors, which is very close to the BCH code (15, 7) with 2bit error correction.

Consideration should also be given to what may be the shortestnon-trivial CPC, which can be addressed without cyclic codeconstruction, but also with no error correction capabilities.

Example 3. For n=3 the communication system may only have 23=8 binarywords of length 3 which have 4 different cyclic permutable codewordsc₁=1, c₂=(1, 1, 0), c₃=(1, 0, 0), c₄=0 allowing the transmitter toencode B=2 bits of information with no error correction. However, thiscode allows a TO estimation, as well as a CIR estimation. By omitting c₁and c₄, i.e., by dropping one bit of information, an estimate can beobtained with the DiZeT oversampling decoder for any possible CFO.

While specific codes and encoder and decoder implementations aredescribed above, communication systems in accordance with variousembodiments of the invention can use any of a variety of codes that arerotation invariant and/or provide bit error correction capabilities asappropriate to the requirements of specific applications in accordancewith various embodiments of the invention. The simulation of specificcommunication systems in accordance with a number of embodiments of theinvention is discussed further below.

Simulations

Monte-Carlo simulations of communication systems in accordance withvarious embodiments of the invention were performed using MatLab 2017athe bit-error-rate (BER) over averaged E_(b)/N₀ and rSNR under variouschannel settings and block lengths K. The transmit and receive time wasin all schemes N=K+L over which the CIR h of length L was assumed to bestatic, see FIG. 11. Here, we schematic illustrate the usage of the Nsample resources of the BMOCZ, Binary PPM, QPSK with delta Pilot,OFDM-IM of order 3 and final OFDM-Differential PSK, utilizing twoconsecutive OFDM symbols with CP1 and CP2 cyclic prefix. The blue markedarea gives the data samples, where the black samples denote the actualactive samples for the data. Note, a Guard or CP requires always atleast L−1 samples, where a Pilot requires L samples. Therefore, theenergy per bit was E_(b)=B/N with B=∥m∥₁ which is the inverse of thespectral efficiency ρ=N/B. In each simulation run, the CIR coefficientswere redrawn according to the channel statistic given by (3) with decayexponent p≤1 and support s. The CFO ϕ was drawn uniformly from [0, 2π)and the timing-offset τ₀ uniformly form [0, N]. Furthermore, theadditive distortion was also drawn from an i.i.d. Gaussian distribution

(0, N₀), which was scaled to obtain various received SNR and E_(b)/N₀values, see FIG. 12A. Here the oversampled BMOCZ-ACPC encoded scheme inpurple filled circles only loses 4 dB in performance compared to the BCHencoded BMOCZ in the low to mediocre SNR regimes. The fractional CFO canbe compensated in high SNR regime by increasing the oversampling factor,see gray squared line,

Due to the embedded error correcting and a complete failure if a wrongCPC codeword is detected, the BER and BLER (block-error-rate) are almostidentical over received SNR for BMOCZ-ACPC, see purple filled circleline in FIG. 12B.

Of course, it would be also possible to detect a wrong decoding andrequest a retransmit.

Multiple Receive Antennas

For a receiver with M antennas, receive antenna diversity can beexploited, since each antenna can receive the transmit signal over anindependent CIR realization (best case). Due to the short wavelength inthe mmWave band large antenna arrays with λ/2 spacing can be easilyinstalled on small devices. It was assumed in all simulations:

-   -   The B information bits m∈        are drawn uniformly.    -   All signals arrive with the same timing-offset τ₀ at the M        receive antennas (dense antenna array, fixed relative antenna        positions (no movements)).    -   The clock-rate for all M antennas is identical, hence all        received signals have the same CFO.    -   The maximal CIR length L, sparsity level S, and PDP p are the        same for all antennas.    -   Each received signal experiences an independent noise and CIR        realization, with sparsity pattern s_(m)∈{0, 1}^(L) for        |supp(s_(m))|=S and h_(m)∈        ^(L) where h_(m,l)=        (0, s_(m,l)p^(l)).        The DiZeT decoder (44) for BMOCZ without TO and CFO, can be        implemented in a straight forward manner relative to a        single-input-multiple-output SIMO antenna system:

$\begin{matrix}{{\overset{\hat{}}{m}}_{k} = \left\{ {\begin{matrix}{1,} & {{\sum\limits_{m = 1}^{M}\;{{{y_{m}{\overset{\sim}{D}}_{\underset{\_}{R}}F^{*}}}^{2}}_{Q{({k - 1})}}} < {R^{K - 2}{\sum\limits_{m = 1}^{M}{{y_{m}{\overset{\sim}{D}}_{{\underset{\_}{R}}^{- 1}}F^{*}}}_{Q{({k - 1})}}^{2}}}} \\{0,} & {else}\end{matrix}.} \right.} & (72)\end{matrix}$

CFO Estimation for Multiple Receive Antennas

In many embodiments, the fractional CFO is estimated by (48) for Mreceived signals y_(m)

$\begin{matrix}{\overset{\hat{}}{q} = {\arg{\min\limits_{q \in {\lbrack Q\rbrack}}{\sum\limits_{k = 1}^{K}{\min{\left\{ {{\sum\limits_{m = 1}^{M}{{{\overset{\sim}{y}}_{m}{\overset{\sim}{D}}_{R}F_{QK}}}_{{Qk} \oplus_{\overset{\sim}{N}}q}},{\sum\limits_{m = 1}^{M}{{\overset{\_}{{\overset{\sim}{y}}_{m}^{-}}{\overset{\sim}{D}}_{R}F_{QK}}}_{{Qk} \oplus_{\overset{\sim}{N}}q}}} \right\}.}}}}}} & (73)\end{matrix}$

The effect of different coding rates on BER and block error rate (BLER)for the BMOCZ-ACPC-(K, B) scheme with K=31 transmitted zeros and channellengths L=16, 32 with flat power profile p=1 is shown in FIGS. 12A and12B. If the coding rate is decreased to B/K=6/31 ≃1/5, allowing up to 5bit error corrections, a BLER of 10⁻¹ can be achieved at almost 6 dBreceived SNR, which is 6 dB better as for the coding rate 16/31 ≃1/2.Hence, if power is an issue, the coding rate can be decreasedaccordingly to approach a low SNR regime, at the cost of data rate. Itcan be seen that for a very sparse and fast decaying power delayprofile, the performance loss will be 1-4 dB, especially if the channellength becomes in the order of the signal length.

Timing and Effective Channel Length Estimation

A random integer timing offset can be chosen in τ₀∈{0, 1, 2 . . . , N−1}in FIG. 13 and with an exponential power delay profile exponent p=0.98and dominant LOS path. In the CPIS algorithm (i.e. Algorithm 1 from FIG.6) and CIRenergy algorithm (i.e. algorithm 2 from FIG. 8) the sum of allreceived antenna samples is used in combination with the sum of thenoise powers

$\begin{matrix}{{r_{n}}^{2} = {{\sum\limits_{m = 1}^{M}{{r_{m,n}}^{2}\mspace{14mu}{and}\mspace{14mu}\sigma^{2}}} = {{MN}_{0}.}}} & (74)\end{matrix}$

Indeed, if the CIR length is larger with exponential decay, a wrong TOestimation yields to less performance degradation as for shorterlengths, since the last channel tap will be much smaller in averagepower. A simulation can be performed by choosing randomly for eachsimulation, consisting of D different noise powers N₀.

Each binary plain message m will result in a codeword c∈

₂ ^(K), which corresponds to a normalized BMOCZ symbol x∈

^(K+1). The CIR can be normalized at the transmitter to {tilde over(h)}=h/

[∥h∥²], where the average energy of the CIR is given by (36) as theexpected power delay profile in s

$\begin{matrix}{E_{S,h} = {{{\mathbb{E}}\left\lbrack {h}^{2} \right\rbrack} = {\sum\limits_{l = 0}^{L - 1}{s_{l}{p^{l}.}}}}} & (75)\end{matrix}$

For each selected sparsity pattern of the CIR, normalized CIR energy isobtained if selecting random channel taps by the law of large numbers.By averaging over the sparsity pattern, this can result in a largedeviation of the CIR energy and would require many more simulations,therefore the average power was calculated with knowledge of thesparsity patterns, i.e., by knowing the support realization.

Comparison to Noncoherent Schemes

First, a comparison is performed without TO and CFO between BMOCZ andOFDM and pilot based schemes, in FIGS. 14A and 14B with a single antennawhere the CIR is generated from the simulation framework Quadriga andfinally in FIGS. 15A and 15B with multiple-receive antennas.

Pilot and SC-FDE with QPSK. A pilot impulse

$\sqrt{\frac{E}{2}}\delta_{0}$

of length P=L can be used to determine the CIR and K+1−L symbols totransmit data via QPSK in a single-carrier (SC) modulation by afrequency-domain-equalization (FDE). Applying FDE, QPSK or QAM modulatedOFDM subcarriers K+1−L can be decoded. The energy E is split evenlybetween the pilots and data symbols, which can result in better BERperformance for high SNR, see FIG. 15

OFDM-Index-Modulation (IM). A comparison can also be performed withrespect to: OFDM-IM with Q=1 and Q=4 active subcarriers out ofK_(IM)=K+1 and to OFDM-Group-Index-Modulation (GIM) with Q=1 activesubcarriers in each group of size C=4. To obtain an OFDM symbol acyclic-prefix is added which requires K_(IM)≥L. For OFDM systems, theCFO will result in a circular shift of the K_(IM) subcarriers and hencecreate the same confusion as for BMOCZ. The only difference is, thatOFDM operates only on the unit circle, whereas BMOCZ operates on twocircles inside and outside the unit circle. Note, in OFDM-IM a cyclicpermutable code is not applicable, since the information for examplewith Q=1 is a cyclic shift, which is exactly what the CFO introduces.For more active subcarriers Q>1 and grouping the carriers in groups, ICIfree IM schemes can be deployed. A group size of G=4 seems to performthe best for OFDM-IM. However, this will require L<<K which is not theproposed regime for MOCZ.

OFDM-Differential-Phase-Shift-keying (DPSK). Two successive OFDM blockscan be used to encode differentially the bits via Q-PSK over K_(dif)subcarriers. To ensure the same transmit and receive lengths as forBMOCZ the BMOCZ symbol length K+1 can be split in two OFDM symbols withcyclic prefix x_(CP) ⁽¹⁾ and x_(CP) ⁽²⁾ of equal length N_(dif)=N/2,where N=K+L is chosen to be even. Furthermore, to include a CP of lengthL−1 in each OFDM block, it is required that N_(dif)=(K+L)/2≥2L−1resulting in the requirement. K≥3L−2. If L is even and K=nL for some2<n∈

it is possible to obtain K_(dif)=N_(dif)−L+1=(n−1)L/2+1 subcarriers ineach OFDM block. The shortest transmission time is then given for even Lwith a=3 by N=4L, resulting in K_(dif)=L+1 subcarriers. Modulating themwith Q-PSK allows to transmit. (L+1) log Q bits differentially. To matchthe spectral-efficiency of BMOCZ as best as possible, it is possible toselect Q=8 to encode 3 bits per subcarrier and henceB=(L+1)3=(K/3+1)3=K+3 message bits, which is 3 bits more than BMOCZ.

The encoding of the DPSK can be done relative to the first. OFDM blocki=1, which will transmit PSK constellation points as with phase zeros_(k) ⁽¹⁾=1 respectively data phases s_(k) ⁽²⁾=e^(j2πq) ^(k) ^(/Q) withq_(k)=bi2de(m_((k-1)log(Q)+1), . . . , m_(k log(Q))) for k=1, . . . ,K_(dif). Hence, in time domain, it is possible to obtain

x=(x _(CP) ⁽¹⁾ ,x _(CP) ⁽²⁾),x _(CP) ^((i))=(CP ^((i)) ,x ^((i))),CP^((i))=(x _(N) _(dif) _(−L+1) ^((i)) , . . . ,x _(N) _(dif) ⁻¹ ^((i))),x^((i)) =F*s ^((i))  (76)

for i=1, 2. Here x will be also normalized.

After removing the CP at the receiver the received data symbols infrequency domain via the mth antenna for the kth carrier is given by

R _(m,k) ^((i)) =H _(m,k) s _(k) ^((i)) +W _(m,k) ^((i))  (77)

where it is possible to consider {W_(m,k) ^((i))} as independentcircularly symmetric Gaussian random variables and H_(m,k)∈

the channel coefficient of the kth subcarrier. Hence, each subcarriercan be seen as a Rayleigh flat fading channel and the decision variablecan be used for a hard-decoding of M antennas

$\begin{matrix}{{\hat{q}}_{k} = {{\underset{q \in {Q}}{\arg\;\min}{{{\frac{1}{M}{\sum\limits_{m = 1}^{M}{R_{m,k}^{(2)}\overset{\_}{R_{m,k}^{(1)}}}}} - e^{j\frac{2\;\pi\; q}{Q}}}}^{2}} = {{int}\left( {\frac{Q}{2\;\pi}{\angle\left( {\frac{1}{M}{\sum\limits_{m = 1}^{M}{R_{m,k}^{(2)}\overset{\_}{R_{m,k}^{(1)}}}}} \right)}{mod}\; Q} \right)}}} & (78)\end{matrix}$

Here int(⋅) rounds to the nearest integer. It is possible to ignore herea possible weighting by knowledge of SNR.

Simulations with Quadriga Channel Simulator

Version 2.0 of the Quadriga channel simulator was used to generaterandom CIRs for the Berlin outdoor scenario (“BERLIN__UMa_NLOS”), withNLOS at a carrier frequency f_(c)=4 Ghz and bandwidth W=150 Mhz, seeFIG. 14. In the simulation, the transmitter and receiver are stationaryusing omnidirectional antennas (λ/2). The transmitter might be a basestation mounted at 10m altitude and the receiver might be a ground userwith ground distance 20m. The LOS distance is then ≈22m.

Computationally Complexity

Since the Q times oversamnpled DiZeT decoder is realized by the QK-pointIDFT, computationally complexity an be reduced for Mersenne primes K, ifthe chosen oversampling factor Q is relatively prime to K. Since K isprime, Q can be chosen to be not a multiple of K. Since N=K+L, L≠mK needonly be chosen for m∈

so that Q=N is relatively prime to K, i.e., only have one as a commondivisor. In this case the QK-DFT can be efficiently computed by theprime-factor algorithm (PFA), which is as efficient at the FFT forlengths of powers of two. During simulation in Matlab, a speedimprovement by a factor of 10 in the oversampled DiZeT decoder wasobserved for K=31 when switching from L=31 to L=32.

As can readily be appreciated, the simulations described above form abasis for implementation of communication systems in accordance withvarious embodiments of the invention using appropriate programmableand/or custom integrated circuits. While specific simulations aredescribed above, any of a variety of simulations and testing processingcan be performed in order to finalize a transmitter and/or receiverdesign as appropriate to the requirements of specific applications inaccordance with various embodiments of the invention.

Although the present invention has been described in certain specificaspects, many additional modifications and variations would be apparentto those skilled in the art. It is therefore to be understood that thepresent invention can be practiced otherwise than specifically describedincluding transmitters and receivers that communicate via any of avariety of communication modalities using MOZ without departing from thescope and spirit of the present invention. Thus, embodiments of thepresent invention should be considered in all respects as illustrativeand not restrictive. Accordingly, the scope of the invention should bedetermined not by the embodiments illustrated, but by the appendedclaims and their equivalents.

1.-20. (canceled)
 21. A receiver, comprising: a demodulator configuredto down convert and sample a received continuous-time signal to obtain areceived discrete-time baseband signal and over sample the receiveddiscrete-time baseband signal, where the received discrete-time basebandsignal includes at least one of a timing offset (TO) and a carrierfrequency offset (CFO); a decoder configured to decode a plurality ofdecoded information bits from the received discrete-time baseband signalby: determining an estimated TO for the received discrete-time basebandsignal to identify a received symbol; determining a plurality of zerosof a z-transform of the received symbol; identifying zeros from theplurality of zeros that encode received bits by identifying andcorrecting a fractional rotation in the plurality of zeros resultingfrom the CFO; and decoding the plurality of decoded information bitsbased upon a plurality of received bits using a cycling register code(CRC).
 22. The receiver of claim 21, wherein the receiver receives thecontinuous-time transmitted signal over a multipath channel.
 23. Thereceiver of claim 21, wherein the z-transform of the discrete-timebaseband signal comprises a zero for each of a plurality of encodedbits.
 24. The receiver of claim 21, wherein each zero in the z-transformof the discrete-time baseband signal is limited to being one of a set ofconjugate-reciprocal pairs of zeros.
 25. The receiver of claim 24,wherein: each conjugate reciprocal pair of zeros in the set ofconjugate-reciprocal pairs of zeros comprises: an outer zero having afirst radius that is greater than one; and an inner zero having a radiusthat is the reciprocal of the first radius; where the inner and outerzero have phases that are the same phase; and the radii of the outerzeros in each pair of zeros in the set of conjugate-reciprocal pairs ofzeros in each pair of zeros in the set of conjugate-reciprocal pairs ofzeros are evenly spaced over one complete revolution.
 26. The receiverof claim 21, wherein the cycling register code is a cyclicallypermutable code (CPC).
 27. The receiver of claim 26, wherein the CPC isextracted from a Bose Chaudhuri Hocguenghem (BCH) code.
 28. The receiverof claim 26, wherein the CPC is extracted from a primitive BCH code. 29.The receiver of claim 26, wherein the CPC has a code length that is aMersenne prime.
 30. The receiver of claim 21, wherein the CRC isgenerated by an inner code and an outer code which are combined in anon-linear fashion.
 31. The receiver of claim 30, wherein the outer codeis an outer cycling register code having a lower code rate than theinner code.
 32. The receiver of claim 30, wherein the outer code is acyclically permutable code (CPC).
 33. The receiver of claim 30, whereinthe CRC is an affine CPC (ACPC) code.
 34. The receiver of claim 33,wherein the ACPC is characterized by being attainable using a cyclicinner code having codewords of an inner codeword length, which is affinetranslated by a given binary word of the inner codeword.
 35. Thereceiver of claim 21, wherein the decoder is configured to determine theestimated TO by measuring energy over an expected symbol length with asliding window in the sampled signal.
 36. The receiver of claim 21,wherein the decoder is configured to measure energy over an expectedsymbol length by convolving samples with a universal Huffman sequence ofthe expected symbol length comprising two impulses at the beginning andthe end of the expected symbol length.
 37. The receiver of claim 21,wherein the decoder is configured to determine the estimated TO byidentifying a set of three energy peaks that yield a maximum energy sumover an expected symbol length.
 38. The receiver of claim 21, whereinthe decoder is configured to determine a most likely set of zeros forthe z-transform of the discrete-time baseband signal used to generatethe transmitted signal based upon the received symbol.
 39. The receiverof claim 21, wherein the decoder is configured to determine theplurality of received bits by performing a weighted comparison ofsamples of the z-transform of the received symbol with each zero in aset of zeros.
 40. The receiver of claim 21, wherein the receivercomprises a plurality of receive antennas and the decoder determines theplurality of decoded information bits by combining values derived fromthe samples of a plurality of continuous-time signals received by theplurality of receive antennas to perform decoding.